Higher-Dimensional Knots According to Michel Kervaire Higher-Dimensional Techniques in This Fascinating Area

Series of Lectures in Mathematics

Françoise Michel Françoise Michel and Claude Weber Claude Weber Françoise Michel Higher-Dimensional Knots Claude Weber According to Michel Kervaire

Michel Kervaire wrote six papers on higher-dimensional knots which can be considered fundamental to the development of the theory. They are not

only of historical interest but naturally introduce to some of the essential Higher-Dimensional Knots According to Michel Kervaire Higher-Dimensional techniques in this fascinating area.

This book is written to provide graduate students with the basic concepts Knots According necessary to read texts in higher-dimensional knot theory and its relations with singularities. The first chapters are devoted to a presentation of Pontrjagin’s construction, surgery and the work of Kervaire and Milnor on to Michel Kervaire homotopy spheres. We pursue with Kervaire’s fundamental work on the group of a knot, knot modules and knot cobordism. We add developments due to Levine. Tools (like open books, handlebodies, plumbings, …) often used but hard to find in original articles are presented in appendices. We conclude with a description of the Kervaire invariant and the consequences of the Hill– Hopkins–Ravenel results in knot theory.

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Previously published in this series (for a complete listing see our homepage at www.ems-ph.org): Sergey V. Matveev, Lectures on Algebraic Topology Joseph C. Várilly, An Introduction to Noncommutative Geometry Reto Müller, Differential Harnack Inequalities and the Ricci Flow Eustasio del Barrio, Paul Deheuvels and Sara van de Geer, Lectures on Empirical Processes Iskander A. Taimanov, Lectures on Differential Geometry Martin J. Mohlenkamp and María Cristina Pereyra, Wavelets, Their Friends, and What They Can Do for You Stanley E. Payne and Joseph A. Thas, Finite Generalized Quadrangles Masoud Khalkhali, Basic Noncommutative Geometry Helge Holden, Kenneth H. Karlsen, Knut-Andreas Lie and Nils Henrik Risebro, Splitting Methods for Partial Differential Equations with Rough Solutions Koichiro Harada, “Moonshine” of Finite Groups Yurii A. Neretin, Lectures on Gaussian Integral Operators and Classical Groups Damien Calaque and Carlo A. Rossi, Lectures on Duflo Isomorphisms in Lie Algebra and Complex Geometry Claudio Carmeli, Lauren Caston and Rita Fioresi, Mathematical Foundations of Supersymmetry Hans Triebel, Faber Systems and Their Use in Sampling, Discrepancy, Numerical Integration Koen Thas, A Course on Elation Quadrangles Benoît Grébert and Thomas Kappeler, The Defocusing NLS Equation and Its Normal Form Armen Sergeev, Lectures on Universal Teichmüller Space Matthias Aschenbrenner, Stefan Friedl and Henry Wilton, 3-Manifold Groups Hans Triebel, Tempered Homogeneous Function Spaces Kathrin Bringmann, Yann Bugeaud, Titus Hilberdink and Jürgen Sander, Four Faces of Number Theory Alberto Cavicchioli, Friedrich Hegenbarth and Dušan Repovš, Higher-Dimensional Generalized Manifolds: Surgery and Constructions Davide Barilari, Ugo Boscain and Mario Sigalotti, Geometry, Analysis and Dynamics on sub- Riemannian Manifolds, Volume I Davide Barilari, Ugo Boscain and Mario Sigalotti, Geometry, Analysis and Dynamics on sub- Riemannian Manifolds, Volume II Dynamics Done with Your Bare Hands, Françoise Dal’Bo, François Ledrappier and Amie Wilkinson, (Eds.) Hans Triebel, PDE Models for Chemotaxis and Hydrodynamics in Supercritical Function Spaces Françoise Michel Claude Weber

Higher-Dimensional Knots According to Michel Kervaire Authors:

Françoise Michel Claude Weber Institut de Mathématiques de Toulouse Section de Mathématiques Université Paul Sabatier Université de Genève 118 route de Narbonne 2-4 rue de Lièvre, C.P. 64 31062 Toulouse Cedex 9 1211 Genève 4 France Switzerland E-mail: [email protected] E-mail: [email protected]

2010 Mathematics Subject Classification: 57Q45, 57R65, 32S55

Key words: Knots in high dimensions, homotopy spheres, complex singularities

ISBN 978-3-03719-180-4

The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography, and the detailed bibliographic data are available on the Internet at http://www.helveticat.ch.

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Typeset using the authors’ TEX files: Alison Durham, Manchester, UK Printing and binding: Beltz Bad Langensalza GmbH, Bad Langensalza, Germany ∞ Printed on acid free paper 9 8 7 6 5 4 3 2 1 Preface

The aim of this book is to present Kervaire’s work on differentiable knots in higher dimensions in codimension q = 2. As explained in our introduction (see Section 1.7), this book is written to make the reading of papers by Michel Kervaire and Jerome Levine easier. Thus we hope to communicate our enthusiasm for higher-dimensional knot theory. In order to appreciate the importance of Kervaire’s contribution, we describe in Chapters2 to4 what was, at the time, the situation in differential topology and in knot theory in codimension q ≥ 3. In Chapter5, we expose Kervaire’s characterization of the fundamental group of a knot complement. In Chapter6, we explain Kervaire and Levine’s work on knot modules. In Chapter7, we detail Kervaire’s construction of the “simple knots” classified by Jerome Levine. Chapter8 summarizes Kervaire and Levine’s results on knot cobordism. In Chapter9, we apply higher-dimensional knot theory to singularities of complex hypersurfaces. AppendixesA toD are devoted to a discussion of some basic concepts, known to the experts: signs, Seifert hypersurfaces, open book decompositions and handlebodies. In AppendixE, we conclude with an exposition of the results of Hill–Hopkins– Ravenel on the Kervaire invariant problem and its consequences for knot theory in codimension 2. The point of view adopted here is somewhat pseudo-historical. When we make explicit Kervaire’s work we try to follow him closely, in order to retain some of the flavor of the original texts. When necessary we add further contributions, often due to Levine. We also propose developments that occurred later. We thank Peter Landweber for patiently correcting our English. The figures in AppendixF are due to Cam Van Quach. We thank her for her friendly collaboration.

Françoise Michel Claude Weber Geneva, September 2016

Contents

Preface...... v

1 Introduction...... 1 1.1 Kervaire’s six papers on knot theory...... 1 1.2 A brief description of the content of the book...... 2 1.3 What is a knot?...... 3 1.4 Knots in the early 1960s...... 4 1.5 Higher-dimensional knots and homotopy spheres...... 4 1.6 Links and singularities...... 5 1.7 Final remarks...... 6 1.8 Conventions and notation...... 7

2 Some tools of diﬀerential topology...... 9 2.1 Surgery from an elementary point of view...... 9 2.2 Vector bundles and parallelizability...... 10 2.3 The Pontrjagin method and the J-homomorphism...... 11

3 The Kervaire–Milnor study of homotopy spheres...... 15 3.1 Homotopy spheres...... 15 3.2 The groups Θn and bPn+1 ...... 15 3.3 The Kervaire invariant...... 20 3.4 The groups Pn+1 ...... 22 3.5 The Kervaire manifold...... 24

4 Diﬀerentiable knots in codimension ≥3...... 27 4.1 On the isotopy of knots and links in any codimension...... 27 4.2 Embeddings and isotopies in the stable and metastable ranges.... 28 4.3 Below the metastable range...... 29

5 The fundamental group of a knot complement...... 33 5.1 Homotopy n-spheres embedded in Sn+2 ...... 33 5.2 Necessary conditions to be the fundamental group of a knot complement 35 5.3 Suﬃciency of the conditions if n ≥ 3 ...... 37 5.4 The Kervaire conjecture...... 40 5.5 Groups that satisfy the Kervaire conditions...... 41 viii Contents

6 Knot modules...... 43 6.1 The knot exterior...... 43 6.2 Some algebraic properties of knot modules...... 45 6.3 The qth knot module when q < n/2 ...... 48 6.4 Seifert hypersurfaces...... 50 6.5 Odd-dimensional knots and the Seifert form...... 52 6.6 Even-dimensional knots and the torsion Seifert form...... 56

7 Odd-dimensional simple links...... 61 7.1 q-handlebodies...... 61 7.2 The realization theorem for Seifert matrices...... 62 7.3 Levine’s classiﬁcation of embeddings of handlebodies in codimension 1 65 7.4 Levine’s classiﬁcation of simple odd-dimensional knots...... 67

8 Knot cobordism...... 69 8.1 Deﬁnitions...... 69 8.2 The even-dimensional case...... 69 8.3 The odd-dimensional case...... 71

9 Singularities of complex hypersurfaces...... 75 9.1 The theory of Milnor...... 75 9.2 Algebraic links and Seifert forms...... 78 9.3 Cobordism of algebraic links...... 79 9.4 Examples of Seifert matrices associated to algebraic links...... 80 9.5 Mumford and topological triviality...... 81 9.6 Joins and Pham–Brieskorn singularities...... 83 9.7 The paper by Kauffman [56]...... 86 9.8 The paper by Kauffman and Neumann [57]...... 87 9.9 Kauffman–Neumann’s construction when both links are the binding of an open book decomposition...... 88

A Linking numbers and signs...... 91 A.1 The boundary of an oriented manifold...... 91 A.2 Linking numbers...... 92

B Existence of Seifert hypersurfaces...... 93

C Open book decompositions...... 95 C.1 Open books...... 95 C.2 Browder’s lemma 2...... 96 Contents ix

D Handlebodies and plumbings...... 99 D.1 Bouquets of spheres and handlebodies...... 99 D.2 Parallelizable handlebodies...... 101 D.3 m-dimensional spherical links in S2m+1 ...... 103 D.4 Plumbing...... 104

E Homotopy spheres embedded in codimension 2 and the Kervaire–Arf–- Robertello–Levine invariant...... 109 E.1 Which homotopy spheres can be embedded in codimension 2?... 109 E.2 The Kervaire–Arf–Robertello–Levine invariant...... 111 E.3 The Hill–Hopkins–Ravenel result and its inﬂuence on the KARL invariant...... 112

F Figures...... 115

Bibliography...... 121

Index of Notation...... 129

Index of Quoted Theorems...... 131

Index of Terminology...... 133

Introduction

1.1 Kervaire’s six papers on knot theory

Michel Kervaire wrote six papers on knots: [63], [64], [66], [45], [46], and [69]. The first one is both an account of Kervaire’s talk at a symposium held at Princeton in spring 1963 in honor of Marston Morse and a report of discussions between several participants in the symposium about Kervaire’s results. Reading this paper is really fascinating, since one sees higher-dimensional knot theory emerging. The subject is the fundamental group of knots in higher dimensions. The second one is the written thesis that Kervaire presented in Paris in June 1964. In fact Kervaire had already obtained a PhD in Zurich under Heinz Hopf in 1955, but in 1964 he applied for a position in France and, at that time, a French thesis was compulsory. In the end, the appointment did not materialize. But the thesis text remains as an article published in the Bulletin de la S.M.F. ([64]). As this article is the main reason for writing this text, we name it Kervaire’s Paris paper. It is the most important one that Kervaire wrote on knot theory. It can be considered to be the foundational text on knots in higher dimensions, together with contemporary papers by Jerome Levine ([77], [78], and [79]). One should also add to the list the Hirsch–Neuwirth paper [50], which seems to be a development of discussions held during the Morse symposium. To briefly present the subject of Kervaire’s Paris paper, we need a few definitions. A knot Kn ⊂ Sn+2 is the image of a differentiable embedding of an n-dimensional homotopy sphere in Sn+2. Its exterior E(K) is the complement of an open tubular neighborhood. The exterior has the homology of the circle S1 by Alexander duality. In short, the subject is the determination of the first homotopy group πq(E(K)), which 1 is different from πq(S ). Later, Kervaire complained that he had to rush to complete this Paris paper in due time and he had doubts about the quality of its redaction. In fact we find this article to be well written. The exposition is concise and clear, typically in Kervaire’s style. The pace is slow in parts that present a difficulty and fast when things are obvious. Now, what was obvious to Kervaire in spring 1964? Among other things, clearly Pontrjagin’s construction and surgery. As these techniques are possibly not so well known to a reader fifty years later, we devote Chapters2 and3 of this text to a presentation of these matters. The last chapter of Kervaire’s Paris paper is a first attempt at understanding the cobordism of knots in higher dimensions. It contains a complete proof that an even- 2 1 Introduction dimensional knot is always cobordant to a trivial knot. For odd-dimensional knots, the subject has a strong algebraic flavor, much related to the isometry groups of quadratic forms and to algebraic number theory. Kervaire liked this algebraic aspect and devoted his third paper [66] to it. Levine spent the first months of 1977 in Geneva and he gave wonderful lectures on many aspects of knot theory. His presence had a deep influence on several members of the audience, including the two authors of this text! Under his initiative a meeting was organized by Kervaire in Les Plans-sur-Bex in March 1977. The proceedings are recorded in the Springer Lecture Notes volume 685, edited by Jean- Claude Hausmann. The irony of history is that it was precisely at this time that William Thurston made his first announcements, which soon completely shattered classical knot theory and 3-manifolds. See Cameron Gordon’s paper [38, p. 44] in the proceedings. The meeting renewed Kervaire’s interest in knot theory, and in the following months he wrote his last three papers on the subject.

1.2 A brief description of the content of the book

As promised, we devote Chapters2 and3 to some background in differential topology, mainly vector bundles, Pontrjagin’s construction, and surgery, culminating with Kervaire–Milnor. We felt it necessary to devote Chapter4 to knots in codimension ≥3. Its reading is optional. One reason to do so is that the subject was flourishing at the time, thanks to the efforts of André Haefliger and Levine. It is remarkable that Levine was present in both fields. Another reason is that the two subjects are in sharp contrast. Very roughly speaking one could say that it is a matter of fundamental group. In codimension ≥3 the fundamental group of the exterior is always trivial while it is never so in codimension 2. But more must be said. In codimension ≥3 there are no PL knots, as proved by Christopher Zeeman [137]. Hence everything is a matter of comparison between PL and DIFF. This is the essence of Haefliger’s theory of smoothing, written a bit later. On the contrary, in codimension 2 the theories of PL knots and of DIFF knots do not differ much. In Chapter5 we present Kervaire’s determination of the fundamental group of knots in higher dimensions, together with some of the results from his two papers written with Jean-Claude Hausmann about the commutator subgroup and the center of these groups. Some later developments are also presented. Chapter6 exposes Kervaire’s results on the first homotopy group πq(E(K)), which 1 −1 is different from πq(S ). For q ≥ 2 these groups are in fact Z[t, t ]-modules. Indeed, Kervaire undertakes a first study of such modules, later to be called knot modules by Levine. In our presentation, we include several developments due to Levine. 1.3 What is a knot? 3

Up to Kervaire’s Paris paper, most of the effort in knot theory went into the construction of knot invariants. They produce necessary conditions for two knots to be equivalent. In Levine’s paper [78] a change took place. From Dale Trotter’s work it was known that for classical knots, Seifert matrices of equivalent knots are S-equivalent. Levine introduced a class of odd-dimensional knots (he called them simple knots) for which the S-equivalence of the Seifert matrices is both necessary and sufficient for two knots to be isotopic. This is a significant classification result. Indeed, simple knots are already present in Kervaire’s Paris paper, but he did not pursue their study that far. Technically, the success of Levine’s study is largely due to the fact that these knots bound a very special kind of Seifert hypersurface: a (parallelizable) handlebody. In Chapter7 we present Levine’s work on odd- dimensional simple knots. Chapter8 is devoted to higher-dimensional knot cobordism. Levine reduced the determination of these groups to an algebraic problem. A key step in the argument rests on the fact that each knot is cobordant to a simple knot. In knot theory, the handlebodies one deals with are parallelizable and their boundary is a homotopy sphere. If we keep the parallelizability condition but admit any boundary, the Kervaire–Levine arguments are still valid. This immediately applies to the Milnor fiber of isolated singularities of complex hypersurfaces, as first noticed by Milnor himself and developed by Alan Durfee. Chapter9 is devoted to that matter. It can be considered a prolongation of Kervaire and Levine’s work. AppendixesA toE are of the kind that can be read independently. They provide basics, comments, and variations on subjects treated elsewhere in our book. In AppendixA we give our conventions on signs, which agree with those in Kauffman– Neumann [57]. This allows us to justify the signs for invariants of some basic algebraic links (examples are given at the end of Chapter9). Often, authors do not mention their sign conventions and hence one can find other signs in the literature. In AppendixB we prove the existence of Seifert hypersurfaces in a more general context. In AppendixesC andD, we present basics about open book decompositions and parallelizable handlebodies, which are useful in knot theory. Our aim in AppendixE is to present the beautiful result by Mike Hill, Mike Hopkins, and Doug Ravenel about the Kervaire invariant and to explain how this affects the theory of knots in higher dimensions. It is a spectacular way to conclude this book. A few figures, related to several sections of the book, are available in AppendixF.

1.3 What is a knot?

We use the following definitions. Definition 1.1. An n-link in Sn+q is a compact oriented differential submanifold without boundary Ln ⊂ Sn+q. The integer q ≥ 1 is the codimension of the link. The 4 1 Introduction

n n n+q → n+q n-links L1 and L2 are equivalent if there exists a diffeomorphism f : S S ( n) n n+q such that f L1 = L2 , respecting the orientation of S and of the links. When Ln is a homotopy sphere, we say that Ln⊂Sn+q is an n-knot in codimension q. When Ln is the standard sphere Sn embedded in Sn+2, we say that Ln ⊂ Sn+2 is a standard n-knot. The group of an n-link (resp. n-knot), Ln ⊂ Sn+2, is the fundamental group of Sn+2 \ Ln. At the beginning of Chapter4, we present the well-known proof that if two links are equivalent there always exists a diffeomorphism f : Sn+q → Sn+q that is isotopic to the identity and moves one link to the other. Hence n-links are equivalent if and only if they are isotopic. In general the boundary of a Milnor fiber is not a homotopy sphere. It is a motivation to explain, in Chapter7, how Kervaire and Levine’s works on simple odd-dimensional knots can be generalized to simple links. A link Ln ⊂ Sn+2 in codimension 2 is always the boundary of a (n+1)-dimensional oriented smooth submanifold Fn+1 in Sn+2. We say that Fn+1 is a Seifert hypersurface for Ln ⊂ Sn+2 ( some authors name it Seifert surface even when n ≥ 2).

1.4 Knots in the early 1960s

In the early 1960s, knots and links in S3 (which we call classical knots and links) had been deeply studied. Two objects played a central role in classical knot theory: the group of the knot and the Seifert surfaces. Despite the fact that, at that time, higher-dimensional topology was ﬂourishing, knowledge about higher-dimensional knots and links was very poor. The only interesting examples were obtained by the spinning construction, which goes back to Emil Artin, and by its generalization by Christopher Zeeman, called twist spinning. For example, if G is the group of a classical knot, the spinning produces, for all n ≥ 2, an n-dimensional knot in Sn+2 having G as knot group. At that time, specialists knew that an n-link is always the boundary of an oriented, smooth, (n + 1)-dimensional submanifold of Sn+2. Here we call it a Seifert hypersurface of the link. A Seifert hypersurface is always parallelizable.

1.5 Higher-dimensional knots and homotopy spheres

We recall, in Chapter3, the results of Kervaire and Milnor [67] concerning the group bPn+1 of n-dimensional homotopy spheres, which bound parallelizable manifolds. 1.6 Links and singularities 5

In particular, bPn+1 is trivial when (n + 1) is odd, and is a finite cyclic group when (n + 1) is even. We also recall, in AppendixE (Proposition E.3), that any element of bPn+1 can be embedded in Sn+2. Conversely an n-dimensional homotopy sphere embedded in Sn+1 is an element of bPn+1 because it is always the boundary of a Seifert hypersurface that is parallelizable. These results lead us to define higher-dimensional knots as embeddings of n-dimensional homotopy spheres in Sn+1. Kervaire had another reason to define higher-dimensional knots as embeddings of homotopy spheres. It is easier to construct higher-dimensional knots without having to specify their differentiable structure. When n is even, an n-dimensional knot is always a standard knot because bPn+1 is trivial. If n = 4k − 1, the signature of the intersection form of a Seifert hypersurface determines the differentiable structure of its boundary. The case n = 4k + 1 is treated in AppendixE.

1.6 Links and singularities

The paper [97] by David Mumford marks the beginning of a new era since it puts a light on the role of the topology in studying complex singularities. In Section 9.5 we explain some consequences of the following Mumford result: Theorem 1.2. Let Σ be a normal complex surface. If the boundary L of point P ∈ Σ is simply connected, then P is a regular point of Σ. Mumford also introduces the concept of “plumbing” to describe the topology of a good resolution of a normal singular point of a complex surface. The reader has to be cautious and should not confuse the plumbings obtained as good resolutions of normal germs of complex hypersurfaces and what we call 2-handlebodies. As explained in AppendixD, following William Browder, there is a way to define 2q-plumbings as equivalent to q-handlebodies if q ≥ 4. The connection between higher-dimensional homotopy spheres and isolated singularities of complex hypersurfaces was established in spring 1966. The story is beautifully (and movingly) told by Egbert Brieskorn in [16, pp. 30–52]. Several mathematicians took part in the events: Egbert Brieskorn, Klaus Jänich, Friedrich Hirzebruch, John Milnor, and John Nash. In June 1966 [15], Egbert Brieskorn proved the following theorem, which is a corollary of his thorough study of the now-named Pham–Brieskorn singularities. The proof rests on the work of Frédéric Pham. Theorem 1.3. Let Σ2q−1 be a (2q − 1)-homotopy sphere that bounds a parallelizable manifold. Then there exist (q + 1) integers ai ≥ 2, 0 ≤ i ≤ q, such that the link ⊂ 2q+1 ( ) Σi=q ai Σ2q−1 Lf S associated to f z0,..., zq = i=0 zi is a knot diffeomorphic to . With his book Singular Points of Complex Hypersurfaces [93], Milnor definitively relates the theory of odd-dimensional links to the study of the embedded topology 6 1 Introduction of a germ f : (Cq+1, 0) → (C, 0) with an isolated critical point at the origin in Cq+1. In Chapter9, we recall many important results contained in [93]. In particular we 2q−1 2q+1 explain how Milnor associates a simple fibered link Lf ∈ S to f . Such a link is called an algebraic link. In [81], Levine gives, when q ≥ 2, a classification theorem for simple (2q − 1)- dimensional knots via the Seifert forms. The knot theory of Kervaire–Levine will directly meet the Milnor theory of algebraic links in a paper by Durfee [29]. Indeed, when q ≥ 3, Durfee shows that the classification theorem of Levine also gives a classification theorem for algebraic links (always in terms of Seifert forms). Such results are based on the classification of q-handlebodies, which are defined in AppendixD. The classification of algebraic links associated to germs of surfaces in C3 is still open. In Section 9.6, we present the notion of joins, following Milnor’s paper [88]. Joins appeared to describe the topology of a germ f = g ⊕ h : (Cn+m, 0) → (C, 0) where g : (Cn, 0) → (C, 0) and h : (Cm, 0) → (C, 0) are germs of holomorphic functions with an isolated critical point at the origin and

f (x1,..., xn, y1,..., ym) = g(x1,..., xn) + h(y1,..., ym).

It gives an inductive method to describe the topology of the Pham–Brieskorn singularities. Kauffman, [56], and Kauffman and Neumann, [57], inspired by the topological behavior of links associated to germs of the type f = g ⊕ h, constructed topologically big families of higher-dimensional links by induction on the dimension. They have constructions where they control the Seifert form, and others with fibered links where they control the open book decompositions.

1.7 Final remarks

The aim of this book is to pay tribute to Kervaire and to make his work on knots of higher dimensions easier to read for younger generations of mathematicians. Basically it is a mathematical exposition text. Our purpose is not to write a history of knots in higher dimensions. We apologize for not making a list of all papers in the subject. When we present Kervaire’s work we try to follow him closely, in order to retain some of the ﬂavor of the original texts. When necessary, we add further contributions often due to Levine. We also propose developments that occurred later. With the passing of time, we ﬁnd it important to present in detail results on the fundamental group of the knot complement and on simple odd-dimensional knots (and links). Indeed, (1) the determination of the fundamental group of the knot complement played a key role at the beginning of higher-dimensional knot theory; 1.8 Conventions and notation 7

(2) higher odd-dimensional simple knots can be classified via their relations with handlebodies; on the one hand this classification induces a classification up to cobordism, and on the other hand, it can be easily generalized to links associated to isolated singular points of complex hypersurfaces. We have wondered whether to include Levine’s name in the title of this text. We have decided not to, although he certainly is the cofounder of higher-dimensional knot theory. But Levine pursued his work much beyond these first years, while Kervaire stopped publishing in the subject (too) early. Hence it would have been difficult to find an equilibrium between them. In fact a study of Levine’s work in knot theory should be much longer than this text.

1.8 Conventions and notation

Manifolds and embeddings are C∞. Usually, manifolds are compact and oriented. A manifold is closed if compact without boundary. The boundary of M is written bM, its interior is M˚ , and its closure is M¯ . Let L be a closed oriented submanifold of a closed oriented manifold M. We denote by N(L) a closed tubular neighborhood of L in M and by E(L) the closure of M \ N(L), i.e., E(L) = M \ N˚(L). By definition E(L) is the exterior of L in M. Fibers of vector bundles are vector spaces over the field of real numbers R. The rank of a vector bundle over a connected base is the dimension of its fibers.

Some tools of diﬀerential topology

An extremely useful reference for this chapter and the next one is provided by Antoni Kosinski’s book [72]. Basics about diﬀerentiable manifolds are beautifully presented by Morris Hirsch in [49].

2.1 Surgery from an elementary point of view

Originally, surgery was “just” a way to transform a differentiable manifold into another one as in [91]. Definition 2.1. The (standard) sphere Sn of dimension n is the set of points x = n+1 Í 2 n+1 (x0, x1,..., xn) ∈ R such that xi = 1. The ball B of dimension (n + 1) is the n+1 Í 2 ˚n+1 set of points x = (x0, x1,..., xn) ∈ R such that xi ≤ 1. The open ball B of Í 2 dimension (n + 1) is defined by xi < 1. Consider the product of spheres Sa × Sb, a ≥ 0, b ≥ 0. This manifold is the boundary of Sa ×Bb+1 and of Ba+1 ×Sb. Suppose now that we have (Sa ×Bb+1) ⊂ M 0 where M 0 is a manifold of dimension m = a+b+1. If M 0 has a boundary, we suppose that the embedding is far from bM 0. We consider then M = M 0 \(Sa × B˚b+1) and we construct M 00 = M ∪ (Ba+1 × Sb) with M ∩ (Ba+1 × Sb) = Sa × Sb. Definition 2.2. One says that M 00 is obtained from M 0 by a surgery along Sa × {0}. The manifold M is the common part of M 0 and M 00. Remarks. (1) The process is reversible: M 0 is obtained from M 00 by surgery along {0} × Sb. (2) Note that M 0 is obtained from the common part M by attaching the cell eb+1 = {u} × Bb+1 of dimension b+ 1 (for some u ∈ Sa) and then a cell em of dimension m. (3) Analogously, M 00 is obtained from M by attaching the cell ea+1 = Ba+1 × {v} of dimension a + 1 (for some v ∈ Sb) and then a cell of dimension m. Definition 2.3. We call Sa ×{0} ⊂ M 0 and {0}×Sb ⊂ M 00 the scars of the surgeries. For an example of surgery, see Figures F.4 and F.5. 10 2 Some tools of differential topology

2.2 Vector bundles and parallelizability

In order to apply surgery successfully, we need to know when a sphere that is differentiably embedded in a differentiable manifold has a trivial normal bundle. Here are some answers. Definition 2.4. A vector bundle over a complex Xk is trivial if it is isomorphic to a product bundle. A trivialization is such an isomorphism. Definition 2.5. A manifold is parallelizable if its tangent bundle τM is trivial. Comment. Parallelizable manifolds were introduced and studied by Ernst Stiefel in his thesis [121], written under the direction of Heinz Hopf. Both were looking for conditions satisfied by a manifold when it is the underlying space of a Lie group. A trivialization was called a parallelism by Stiefel. He proved that, among spheres, S1, S3, S7 are parallelizable. The following result was proved by Kervaire [60] and also by Bott–Milnor [14]. A key step is provided by the Bott periodicity theorem [13]. Theorem 2.6. S1, S3, S7 are the only spheres that are parallelizable. Notation. Let r denote the trivial bundle of rank r over an unspecified basis. Definition 2.7. A vector bundle η is stably trivial if η ⊕ r is trivial for some integer r ≥ 0. Proposition 2.8. Let Xk be a complex of dimension k. Let η be a vector bundle over Xk of rank s with k < s. Suppose that η is stably trivial. Then it is trivial. Proof. See [91, Lemma 4]. To be parallelizable is a very strong condition on a manifold, quite difficult to handle. The next condition is easier to work with. Definition 2.9. A manifold M is stably parallelizable if its tangent bundle is stably trivial. Comments. (1) If M is stably parallelizable then τM ⊕1 is trivial, by the proposition above. (2) The same proposition implies that a compact connected manifold Mn with nonempty boundary is stably parallelizable if and only if it is parallelizable, since it has the homotopy type of a complex of dimension n − 1 and since the classification of bundles depends only on the homotopy type of the base space. (3) The standard embedding of Sn in Rn+1 shows that Sn is stably parallelizable for all n. 2.3 The Pontrjagin method and the J-homomorphism 11

(4) More generally, suppose that Mn is embedded in a stably parallelizable manifold with trivial normal bundle. Then Mn is stably parallelizable. Proposition 2.10. Let Mn be a stably parallelizable manifold of dimension n, embedded in a stably parallelizable manifold W N of dimension N with N ≥ 2n + 1. Then the normal bundle ν of M in W is trivial. Proof. SinceW is stably parallelizable, we have τW⊕1 = N+1. Hence τM⊕ν⊕1 = N+1. Since M is stably parallelizable we have ν ⊕ n+1 = N+1. Since the rank of ν is strictly larger than the dimension of M, Proposition 2.8 applies. Comments. (1) In practice, the last proposition is applied to perform surgery on embedded spheres in stably parallelizable manifolds. (2) The proposition is not true in general if N = 2n. See the sections below about the Kervaire invariant and about the groups Pn+1.

2.3 The Pontrjagin method and the J-homomorphism

The Pontrjagin method is also called the Thom–Pontrjagin method or construction. See [104] and [124]. Its importance lies in the fact that it ties together (stably) parallelizable manifolds and (stable) homotopy groups of spheres. Originally, Pontrjagin wished to compute homotopy groups of spheres via differential topology. In the hands of Kervaire and Milnor it provided the crucial link between homotopy groups of spheres and groups of homotopy spheres. (The novice reader should breathe deeply and read the last sentence a second time.) We can see in retrospect that Kervaire’s early work as a graduate student was ideal preparation for a young topologist in the 1950s. Heinz Hopf asked Kervaire to read Pontrjagin’s announcement note [103] and to provide proofs where needed. See [59, bottom p. 220]. Kervaire’s description of the starting point of his thesis is quite premonitory: “Une variété fermée Mk étant plongée dans un espace euclidien Rn+k avec un champ de repères normaux. . . ” (A closed manifold Mk being embedded in a Euclidean space Rn+k with a field of normal frames. . . .) See [59, middle p. 219]. The first chapter of Kervaire’s thesis in Zurich is devoted to a detailed presentation of Pontrjagin’s construction. Here is a short reminder about the Pontrjagin method. We shall need the stable version of it only. Let Mn be a closed stably parallelizable manifold. We do not assume that Mn is connected. Choose an integer m such that m ≥ n + 2. By Whitney’s theorems [134] we can embed Mn in Sn+m and any two embeddings are isotopic. By Section 2.2 its normal bundle is trivial. We choose a trivialization τ of it. More precisely, we choose 12 2 Some tools of differential topology

m sections of the normal bundle e1, e2,..., em such that e1(x), e2(x),..., em(x) are linearly independent for each x ∈ Mn. These sections provide a map ψ : (U; bU) → (Bm; bBm) where U denotes a closed tubular neighborhood of Mn in Sn+m. The sphere Sm is identified with Bm/bBm. Let ψ¯ : U/bU → Bm/bBm = Sm be the induced map. Extend ψ¯ to a map Ψ : Sm+n → Sm by sending the complement of U in Sn+m to the smashed boundary bBm. It is easily verified that the homotopy class of Ψ depends only on (M, F). This is the essence of the Pontrjagin construction. For short we call the couple (Mn ⊂ Sn+m, τ) a framed n-manifold (in fact a manifold in Sn+m with a trivialization (framing) of its normal bundle). Two n framed n-manifolds Mi for i = 0, 1 are framed cobordant if there exists a framed n+1 n+m n+m W ⊂ S × I such that W ∩ S × {i} = Mi consistently with the framings. fr Framed cobordism classes constitute an abelian group Ωn,m under disjoint union. Let us observe that homotopic trivializations give rise to framed cobordant manifolds fr and hence produce the same element in Ωn,m. Theorem 2.11 (Pontrjagin isomorphism theorem). The Pontrjagin construction pro- fr m vides an isomorphism between Ωn,m and πn+m(S ). Proof. From the definition of the homotopy relation, we deduce that framed cobordant submanifolds in Sn+m give rise to homotopic maps Sn+m −→ Sm. Hence we get fr m a homomorphism Ωn,m −→ πn+m(S ). To prove surjectivity, we argue as follows. Let f : Sn+m −→ Sm be a represen- m m tative of some element in πn+m(S ). Let P ∈ S be a regular value of f equipped with a normal framing in Sm. Then, by transversality, f −1(P) = M is a framed submanifold of Sn+m and the Pontrjagin construction applied to M (essentially the smashings) produces a map Sn+m −→ Sm that is homotopic to f . Here now is the proof of injectivity. First remember that a map f : Sn+m −→ Sm is homotopic to a constant if and only if it extends to a map f ∗ : Bn+m+1 −→ Sm. Suppose then that f is obtained by the Pontrjagin construction and that it represents m ∗ m the trivial element of πn+m(S ). Make f transversal to P ∈ S keeping it fixed on Sn+m. Again by transversality the framed submanifold in Sn+m that represents f bounds in Bn+m+1 the framed submanifold ( f ∗)−1(P). By the definition of framed fr cobordism it represents the trivial element of Ωn,m. The proof is clearly valid without stability assumptions. But since we assume that m fr m ≥ n + 2, the groups πn+m(S ) and Ωn,m are independent of m. We denote them by S fr S πn and Ωn . The group πn is the nth stable homotopy group of spheres (also called S the stable nth stem). By Jean-Pierre Serre’s thesis [112], πn is a finite abelian group. In [61] Kervaire gave interpretations of several constructions in the theory of homotopy groups of spheres via the Pontrjagin method. For instance, the method yields an easy description of a family of homomorphisms originally due to Hopf and Whitehead. Let n ≥ 1 be a fixed integer and let m ≥ 1 be some variable integer. 2.3 The Pontrjagin method and the J-homomorphism 13

Consider the nth sphere Sn standardly embedded in Sn+m. Its normal bundle has a standard trivialization and hence (homotopy classes of) diﬀerent trivializations are in natural bijection with πn(SOm). The Pontrjagin construction yields a homomorphism m Jn,m : πn(SOm) → πn+m(S ). If m is large with respect to n (precisely if m ≥ (n+2)), the group πn(SOm) does not depend on m and is denoted by πn(SO). Moreover, the homomorphism Jn,m does not depend on m. S Thus we get a homomorphism Jn : πn(SO) → πn . This is the stable J- homomorphism in dimension n. Just at the right time, Raoul Bott [13] computed the groups πn(SO). His celebrated results, Bott periodicity, are as follows: Theorem 2.12. Let n ≥ 1 be an integer: then,

(1) πn(SO) depends only on the residue class of n mod 8;

(2) πn(SO) is isomorphic to Z/2 if n ≡ 0 or 1 mod 8;

(3) πn(SO) is isomorphic to Z if n ≡ 3 or 7 mod 8;

(4) πn(SO) is equal to 0 in the four other cases n ≡ 2, 4, 5, 6 mod 8. Frank Adams’ important results [4] about the stable J-homomorphism are stated in the next two theorems. The ﬁrst one is indispensable for proving that a homotopy sphere of dimension n is stably parallelizable when (n − 1) is congruent to 0 or 1 mod 8. See Section 3.2.

Theorem 2.13. The homomorphism Jn is injective when πn(SO) is isomorphic to Z/2. S When πn(SO) is isomorphic to Z the image Im Jn ⊂ πn is a ﬁnite cyclic subgroup whose elements are by their very construction easy to describe. Hence it was important to determine what this subgroup is. Milnor–Kervaire in [94] (with the help of Atiyah–Hirzebruch [7] to get rid of a possible factor 2) gave a “lower bound” (i.e., a factor) of its order. Then Adams proved that this “expected value” is the right one. Another possible factor 2 was eliminated by the proof of the Adams conjecture. See [5, pp. 529–532]. The proof of the Adams conjecture also established that Im Jn is a S direct factor of πn . The ﬁnal statement follows:

Theorem 2.14. Let us write n = 4k − 1. Then the order of the image of J4k−1 : π ( ) → πS (B / k) B k 4k−1 SO 4k−1 is equal to the denominator den k 4 where k is the th Bernoulli number, indexed as by Friedrich Hirzebruch in [52].

We recall that this numbering is such that Bk > 0 for every k > 0 and that the sequence begins B1 = 1/6, B2 = 1/30,.... It is fortunate for topologists that this denominator is computable (it is the “easy part” of Bernoulli numbers, much related to von Staudt theorems). See [3] for explicit formulas. 14 2 Some tools of diﬀerential topology

Remark on the history. We have presented one of the early appearances of the J-homomorphism. Many others followed. Here is one of them, also due to Kervaire and Milnor, but never published. Let ν be a very large integer. Let Gν be the set of ν−1 ν homotopy equivalences of the sphere S . Let Oν be the orthogonal group of R . We have a natural inclusion Oν ,→ Gν. Let us take the limit for ν going to inﬁnity. We obtain a map J : O → G that induces a homomorphism πn(O) → πn(G). This S homomorphism is equivalent, via an isomorphism between πn and πn(G), to the S homomorphism Jn : πn(SO) → πn deﬁned above via the Pontrjagin construction. It is the reason why the map J : O → G is also called the J-homomorphism. 3

The Kervaire–Milnor study of homotopy spheres

In this chapter we oﬀer a summary of Kervaire–Milnor’s paper. The reader interested only in knot theory can safely skip over this chapter.

3.1 Homotopy spheres

Definition 3.1. A differential closed oriented manifold Σn of dimension n is a homotopy sphere if it has the homotopy type of the standard sphere Sn. Comments. (1) Classical results of algebraic topology imply that Σn is a homotopy n n n n sphere if and only if π1(Σ ) = π1(S ) and Hi(Σ ; Z) = Hi(S ; Z) for all i = 0, 1,..., n. (2) When [67] was written it was just known (thanks to Stephen Smale and John Stallings) that a homotopy sphere of dimension n ≥ 5 is in fact homeomorphic to the standard sphere Sn. See [116] and [118]. (3) It is known today that a homotopy sphere of dimension n is homeomorphic to Sn for any value of n. In fact a stronger result is known. Consider a compact and connected topological n-manifold that has the homotopy type of the n-sphere. Then this manifold is homeomorphic to Sn. We could say that “the topological Poincaré conjecture is true in all dimensions”. Stated in this form, the result is due to Newman [101] for n ≥ 5 and to Freedman [36] for n = 4. The proof is classical for n ≤ 2, but it uses the triangulation of surfaces. For n = 3 the proof is tortuous. First apply Moise [96] to triangulate the homotopy sphere. Then smooth it by Munkres [98]. Finally apply Perelman! Proposition 3.2. A homotopy sphere of dimension n ≥ 5 is diffeomorphic to the standard sphere if and only if it bounds a contractible manifold. Proof. This is an immediate consequence of the Smale h-cobordism theorem.

3.2 The groups Θn and bPn+1

We consider the set Λn of oriented homotopy n-spheres up to orientation-preserving diﬀeomorphism. On Λn a composition law is deﬁned, called the connected sum 16 3 The Kervaire–Milnor study of homotopy spheres

and written Σ1]Σ2. See [90] for a comprehensive study of this operation. It is commutative, associative, and has the standard sphere as the zero element. In other words, it is a commutative monoid. By the validity of the topological Poincaré conjecture, Λn classifies the oriented differential structures on the n-sphere up to orientation-preserving diffeomorphism. To avoid confusing readers who go through papers of the 1950s and 1960s, we keep the terminology “homotopy spheres”. n+1 ( ) n Definition 3.3. Let W be an oriented compact n + 1 -manifold and let M1 and n n+1 M2 be two oriented n-manifolds. Then W is an oriented h-cobordism between n n M1 and M2 if n Ý − n − (1) the oriented boundary bW is diffeomorphic to M1 M2 , where the sign denotes the opposite orientation; n n+1 (2) both inclusions Mi ,→ W are homotopy equivalences. Kervaire and Milnor prove that the h-cobordism relation is compatible with the connected sum operation. The quotient of Λn by the h-cobordism equivalence relation is an abelian group written Θn. The inverse of Σ is −Σ. Comments. (1) If n ≥ 5 by Smale’s h-cobordism theorem Θn is in fact isomorphic to Λn. Note that Kervaire–Milnor say explicitly that their paper does not depend on Smale’s results. But of course Smale’s results are needed to interpret Θn in terms of differential structures on Sn. (2) Since homotopy spheres are oriented, a chirality question is present here. A homotopy sphere Σn is achiral (i.e., possesses an orientation-reversing diffeomorphism) if and only if it represents an element of order ≤2 in Θn (assume n , 4). The starting point of [67] is the following result. Theorem 3.4. Any homotopy sphere Σn is stably parallelizable. Rough idea of the proof. We first need a definition. Definition 3.5. A closed connected differentiable manifold Mn is almost parallelizable if Mn \{x} or equivalently Mn \ B˚n is parallelizable. Clearly a homotopy sphere is almost parallelizable. For an almost parallelizable manifold Mn there is one obstruction to stable parallelizability, which is an element of πn−1(SO). By Pontrjagin construction this obstruction lies in the kernel of the J-homomorphism. If πn−1(SO) is isomorphic to Z/2 by Adams’ theorem the J- homomorphism is injective in these dimensions and hence the obstruction vanishes. If πn−1(SO) is isomorphic to Z, a nonzero multiple of the obstruction is equal to the signature of Mn and hence equal to 0 if Mn is a homotopy sphere. 3.2 The groups Θn and bPn+1 17

Comment. In fact the argument proves the following result: An almost parallelizable n-manifold Mn is stably parallelizable if n ≡ 1, 2, 3 mod 4. If n ≡ 0 mod 4 then Mn is stably parallelizable if and only if its signature vanishes. Kervaire and Milnor can then apply the Pontrjagin method to elements of Θn as follows. Let Σn represent an element x of Θn. Embed Σn in Sn+m for m large (m ≥ n + 2). By Proposition 2.10 its normal bundle ν is trivial. We choose a S trivialization of ν. The Pontrjagin method then produces an element fn(x) of πn that does not depend on the choice of Σn to represent x nor on the embedding of Σn in Sn+m. It depends however on the choice of the trivialization of ν. The indeterminacy S lies in the subgroup Im Jn of πn . We obtain therefore a homomorphism n S fn : Θ → πn /Im Jn = Coker Jn. The main objective of [67] is to determine the kernel and cokernel of this homomorphism by surgery. The results of Kervaire–Milnor are the following. First the homomorphism fn is almost always surjective. Theorem 3.6 (Kervaire–Milnor). (1) For n ≡ 0, 1, 3 mod 4 the homomorphism n fn : Θ → Coker Jn is surjective. S (2) For n ≡ 2 mod 4 there is a homomorphism KIn : πn → Z/2 called the Kervaire n invariant such that fn : Θ → Coker Jn is surjective if and only if KIn is the trivial homomorphism. The deﬁnition and properties of the Kervaire invariant are postponed to the next section.

Comment. Usually the J-homomorphism Jn is not surjective. But, since fn is almost S surjective, we can often represent an element of πn by an embedded homotopy sphere, instead of the standard sphere.

The kernel of fn is described by the next proposition. n Proposition 3.7. A homotopy sphere Σ represents an element in the kernel of fn if and only if it bounds a parallelizable manifold. Proof. We argue essentially as in the proof of the Pontrjagin isomorphism theorem. Suppose that Σn bounds a parallelizable manifold Fn+1. Let ϕ : Fn+1 −→ Bn+m+1 be an embedding such that ϕ−1(Sn+m) = Σn for some m ≥ n + 2. Since Fn+1 is parallelizable, the normal bundle of ϕ(Fn+1) is trivial. We choose a trivialization τ. fr n Then (Σn; τ) represents the trivial element in Ωn . Hence Σ represents the trivial m element of the quotient πn+m(S )/Im Jn = Coker Jn for some trivialization of its stable normal bundle. The argument for the opposite implication consists in going backwards along the path we have just followed. 18 3 The Kervaire–Milnor study of homotopy spheres

n n+1 Notation. The kernel of fn : Θ → Coker Jn is denoted by bP by Kervaire– Milnor, who prove the following theorem. Theorem 3.8. The group bPn+1 is a ﬁnite cyclic group. More precisely, (1) it is the trivial group if n + 1 is odd; (2) if n + 1 ≡ 2 mod 4 it is either 0 or Z/2; more precisely, bP4k+2 = Z/2 if and only if KI4k+2 is trivial; (3) if n + 1 = 4k then bPn+1 is nontrivial, with order equal to 2k−2 2k−1 2 (2 − 1) num(4Bk/k).

Comment. The numerator of 4Bk/k is the “hard part” of Bernoulli numbers. It is a product of irregular primes (Kummer’s results about the Fermat conjecture) and tends to inﬁnity at a vertiginous speed. See the impressive “Bernoulli number page” [9]. Summing up, we obtain a Kervaire–Milnor short (indeed, not quite short) exact sequence (where we extend the deﬁnition of KIn to other values of n to be the 0-homomorphism):

n+1 n S 0 → bP → Θ → πn /Im Jn → Im KIn → 0. This exact sequence is valid for all values of n ≥ 1. In fact for n ≤ 6 one has Θn = 0 = bPn+1. This is due to Kervaire–Milnor for n ≥ 4 and Perelman S for n = 3. Moreover, one has πn /Im Jn = 0 = Im KIn for n = 1, 3, 4, 5 and S πn /Im Jn = Z/2 = Im KIn for n = 2, 6. Typically, in these last two dimensions, the S 1 1 3 3 nontrivial element in πn /Im Jn is represented by a framing on S × S and on S × S . Serious affairs begin with n = 7. For us an important fact is expressed in the following remark. It is an immediate consequence of the Pontrjagin method. Remark. A homotopy sphere Σn represents an element of bPn+1 if and only if it bounds a parallelizable manifold (since such a manifold has necessarily a nonempty boundary, “parallelizable” is equivalent to “stably parallelizable”). Comments. (1) A Kervaire–Milnor short exact sequence relates Θn to two objects that are at the heart of mathematics. Both are quite difficult to compute explicitly: S S (i) Coker Jn = πn /Im Jn is the “hard part” of πn . It is known (Adams) that it S can be identified with a direct summand of πn and is essentially accessible via spectral sequences (Adams, Novikov, and many others). (ii) The hard part of Bernoulli numbers is a mysterious subject. Kummer’s results imply that it is a product of irregular primes and that every irregular prime appears at least once in the hard part of some Bk. A look at existing tabulations is impressive. See the impressive “Bernoulli number page” [9]. 3.2 The groups Θn and bPn+1 19

(2) It is known today [21] that a Kervaire–Milnor short exact sequence splits at Θn. Kervaire tried to prove this in 1961–62 (letter to André Haeﬂiger dated 13 Jan 1962). S n (3) From existing tables of πn and Bk (e.g., on the Web), one can determine Θ without pain for roughly n ≤ 60. Very likely, specialists in the 2-primary component of Coker Jn can improve the computations to n ≤ 100 (and maybe more?). See [107] for n ≤ 60.

The rival group Γn. Historically (and conceptually) two groups are in competition: Θn and Γn. The second one was first defined by Thom (1958) in his program to smooth PL manifolds. Its definition is lucidly given by Milnor in [90]. See also [92]. Here it is: Milnor says that an oriented differential structure Σn on the n-sphere is a twisted n-sphere if Σn admits a Morse function with exactly two critical points. From Morse theory, this is equivalent to saying that Σn is obtained by gluing two closed n-balls along their boundary with an orientation-preserving diffeomorphism. Then we consider the subset Γn of Λn, which consists of elements represented by twisted spheres. It is a group (abelian) since there is an isomorphism: Diff+(Sn−1) Γn = . rDiff+(Bn) Here Diff+(Sn−1) denotes the group of orientation-preserving diffeomorphisms of Sn−1 and Diff+(Bn) the group of orientation-preserving diffeomorphisms of the closed ball Bn. The letter r denotes the restriction homomorphism. Today, it is known that Γn = 0 for n ≤ 3 by Smale [115] and Munkres [98] and Γ4 = 0 by Cerf [26]. Of course Θn = 0 for n = 1, 2 and by Perelman Θ3 = 0. Kervaire–Milnor proved that Θ4 = 0. Now Smale’s results imply that Λn = Γn = Θn for n ≥ 5. Hence, finally, Γn = Θn for all values of n. Remarks. (1) For n , 4, every homotopy n-sphere can be obtain by gluing two discs along their boundary. (2) For n = 4, Γ4 = 0 means that every differential structure on the 4-sphere constructed by gluing two closed 4-balls by a diffeomorphism of their boundary is diffeomorphic to the standard differential structure. But it is unknown whether every differential structure on the 4-sphere can be obtained by such a gluing. On the other hand, Θ4 = 0 means that every homotopy 4-sphere is h-cobordant to S4. But we cannot apply Smale’s h-cobordism theorem, which requires that the dimension of the h-cobordism be ≥6. Moreover, it is known today that the differential simply connected h-cobordism theorem is not valid in dimension 5. This is an immediate consequence of two results: 20 3 The Kervaire–Milnor study of homotopy spheres

(1) Simply connected differential 4-manifolds that are homeomorphic are h-cobordant. This follows from Wall in [130]. (2) Many examples of 4-dimensional differentiable manifolds that are homeomorphic but not diffeomorphic are known. The first such examples were constructed by Simon Donaldson. Note that n = 4 is the only dimension for which we do not know whether Λn is a group.

3.3 The Kervaire invariant

πS → Z/ We define the Kervaire invariant KI4k+2 : 4k+2 2 with the help of Pontrjagin’s construction. In this setting KI4k+2 is defined as an obstruction to framed surgery. The fact that the Arf invariant is an obstruction to surgery in dimensions n ≡ 2 mod 4 was first announced by Milnor in [89] where, we think, Arf is quoted for the first time in differential topology. Surprisingly, Kervaire does not cite Arf in [62], but Kervaire–Milnor [67] do. x ∈ Ωfr πS (x) ∈ Z/ Let 4k+2 = 4k+2. The element KI4k+2 2 is the obstruction to represent x by a homotopy sphere. Here are some details. For simplicity assume k ≥ 1. We represent x by a differentiable manifold M4k+2 differentiably embedded in S(4k+2)+m (m ≥ (4k + 2) + 2) with a framing of its normal bundle. By easy surgery (as explained by Milnor in [91]) we can assume that M4k+2 is 2k-connected. The 4k+2 group H2k+1(M ; Z) is a free abelian group of even rank 2r since the intersection 4k+2 form I is alternating and unimodular. By the Hurewicz theorem H2k+1(M ; Z) = 4k+2 4k+2 π2k+1(M ). Since M is simply connected, by Whitney’s elimination of double 4k+2 points [135] every element of H2k+1(M ; Z) can be represented by an embedded sphere S2k+1 ⊂ M4k+2. The normal bundle ν of S2k+1 is stably trivial of rank (2k +1). We summarize the prerequisites in a lemma. See [72, Appendix 1.5] for proofs. Lemma 3.9. (1) Isomorphism classes of oriented vector bundles of rank r over the d+1 sphere S are in a natural bijection with πd(SOr ). This is a special case of Feldbau’s classification theorem. (2) Stably trivial vector bundles of rank (2k + 1) over S2k+1 are in bijection with the kernel K of π2k(SO2k+1) → π2k(SO). (3) ) This kernel is cyclic, generated by the tangent bundle τ of S2k+1. (4) The order of τ in K is equal to 1 if and only if k = 0, 1, 3 (this corresponds to the dimensions of the spheres that are parallelizable). For other values of k the order of τ is equal to 2. 3.3 The Kervaire invariant 21

There are now two cases. (1) Assume that 2k + 1 , 1, 3, 7. In this case ν ∈ K is not necessarily trivial. Let o(ν) ∈ Z/2 be equal to 0 if ν is trivial and to 1 if ν = τ. The correspondence (S2k+1 ⊂ M4k+2) 7→ ν 7→ o(ν) gives rise to a quadratic form

4k+2 q : H2k+1(M ; Z/2) → Z/2.

More precisely, we have the equality in Z/2 : q(y1+y2) = q(y1)+q(y2)+I2(y1, y2). Here I2 denotes the intersection form I reduced mod 2. Since the equality takes place in Z/2, the quadratic form q is not determined by the bilinear form I2.

Definition 3.10. Let (e1, e2, ... , er , f1, f2, ... , fr ) be a symplectic basis of 4k+2 H2k+1(M ; Z). The Arf invariant Arf(q) ∈ Z/2 is defined as r Õ Arf(q) = q(ei)q( fi). i=1 Claim. We have (1) Arf(q) is the only obstruction to transform M4k+2 by framed surgery to a homotopy sphere; (q) x ∈ Ωfr πS (2) Arf depends only on the element 4k+2 = 4k+2 and not on the manifold M4k+2 chosen to represent it. By definition x ∈ Ωfr πS 7→ M4k+2 7→ (q) the correspondence 4k+2 = 4k+2 Arf gives rise to the Kervaire invariant

S → / KI4k+2 : π4k+2 Z 2. (2) Assume that 4k + 2 = 2, 6, 14. In this case the normal bundle ν is trivial. However there is still an obstruction to transform M4k+2 by a framed surgery to a homotopy sphere. It is also given by the Arf invariant of a quadratic form 4k+2 on H2k+1(M , Z/2). The statements of the claim are also valid in this case. For 4k + 2 = 2 this way of reasoning goes back to Pontrjagin [104] who actually S → Z/ proved that KI2 : π2 2 is an isomorphism. The ingredients of the proofs are scattered in [67, Section 8] and in fact already in [62] for 4k + 2 = 10. A comprehensive presentation is in [84, pp. 814] and in Kosinski’s book [72, Section X]. Note. Another approach to the Kervaire invariant as an obstruction to surgery uses immersion theory; see [133]. There are several other deﬁnitions of the Kervaire invariant, more relevant to stable homotopy groups of spheres. William Browder’s 22 3 The Kervaire–Milnor study of homotopy spheres paper [18] relates the Kervaire invariant to the Adams spectral sequence. See also several papers by Edgar Brown. We have merely presented the beginning of a long story. The Kervaire invariant problem is to determine for each n = 4k + 2 whether the homomorphism KIn is trivial or not. An important step in Kervaire’s paper [62] is the proof that KI10 is trivial. In April 2009, Mike Hill, Mike Hopkins, and Doug Ravenel [47] announced that they had solved the Kervaire invariant problem. Soon afterwards, they published a complete proof. Prior to Hill–Hopkins–Ravenel it was known that

(i) KI8k+2 is trivial for k ≥ 1 (Brown–Peterson (1966) in [20]); u (ii) KI4k+2 is trivial if (4k + 2) , 2 − 2 (Browder (1969) in [18]). However,

(1) KI2 is nontrivial (Pontrjagin (1955!) in [104]);

(2) KI6 and KI14 are nontrivial (Kervaire–Milnor (1963) in [67]);

(3) KI30 is nontrivial (Barratt (1967 unpublished) and Mahowald–Tangora (1967) in [86, § 8]);

(4) KI62 is nontrivial (Barratt–Jones–Mahowald (1982) in [8]).

Right after the publication of [67] it was generally conjectured that KI4k+2 is trivial if 4k + 2 , 2, 6, 14. After the publication of Browder’s paper it was conjectured by several algebraic topologists that KI4k+2 is nontrivial in the dimensions left open by Browder, i.e., 4k + 2 = 2u − 2. This was conﬁrmed for 4k + 2 = 30, 62. Mahowald derived important consequences of the second conjecture. The answer provided by Hill–Hopkins–Ravenel is close to the ﬁrst one.

Theorem 3.11 (Hill–Hopkins–Ravenel). The homomorphism KI4k+2 is trivial for all (4k+2) except 2, 6, 14, 30, 62, and perhaps 126.

3.4 The groups Pn+1

Kervaire and Milnor did not actually deﬁne the group Pn+1 in their paper. But they knew everything about it, since, in fact, they compute it. Only the “boundary quotient group” bPn+1 is present. As we shall meet the manifolds involved in the deﬁnition of the group Pn+1 (as Seifert hypersurfaces for knots in higher dimensions) we say a few words about it. Roughly speaking elements of Pn+1 are represented by parallelizable (n + 1)-manifolds which have a homotopy sphere as boundary. 3.4 The groups Pn+1 23

More precisely, let Fn+1 be an (n+1)-dimensional manifold embedded in R(n+1)+m with a trivialization of its normal bundle. We suppose that m ≥ (n+1)+2 and that the boundary bF is a homotopy sphere. An equivalence relation (framed cobordism) between these manifolds is deﬁned as follows: F0 and F1 are framed cobordant if there exists a framed submanifold W n+2 embedded in R(n+1)+m × [0, 1] such that

(n+1)+m W ∩ R × {i} = Fi for i = 0, 1,

bW = F0 ∪ (−F1) ∪ U, where U is an h-cobordism between bF0 and bF1. One of Kervaire–Milnor’s important results is the computation of the groups Pn+1. It goes as follows:

Theorem 3.12.

• Pn+1 = 0 if (n + 1) is odd.

• Pn+1 = Z if (n + 1) is divisible by 4.

• Pn+1 = Z/2 if (n + 1) ≡ 2 mod 4.

We offer some comments on this theorem. The proof that Pn+1 = 0 if (n + 1) is odd is highly nontrivial. It occupies 20 pages in the paper. Here is a description of how the isomorphisms with Z and Z/2 are obtained: For (n + 1) even, let us write n + 1 = 2q. Assume q ≥ 3. By easy surgery arguments (essentially [91]), Kervaire–Milnor prove that every element of P2q can 2q be represented by a (q − 1)-connected manifold, say W . Now Hq(W; Z) = πq(W) is a finitely generated free abelian group. By Whitney’s elimination of double points [135], since q ≥ 3 every element of Hq(W; Z) can be represented by an embedded sphere Sq ,→ W2q. The problem is to determine its normal bundle ν. Since W2q is stably parallelizable, ν is stably trivial but not necessarily trivial. The problem now splits, depending on the parity of q. If q is even, a stably trivial vector bundle η of rank q on the sphere of dimension q is classified by an element o(η) ∈ Z. For the embedded Sq ,→ W2q the element o(ν) that classifies the normal bundle ν is equal to the self-intersection of Sq in W2q. It follows that the intersection bilinear form IW on Hq(W; Z) classifies the normal bundle of embedded spheres representing elements of Hq(W; Z). By Lemma 3.11 and Theorem D.6, the form IW is even. It turns out that the obstruction to obtain 2q a contractible manifold by framed surgery from W is the signature σ(W) of IW . 2q Since IW is even, σ(W) is divisible by 8 and the desired homomorphism P → Z is given by the signature divided by 8. By construction this homomorphism is injective. 24 3 The Kervaire–Milnor study of homotopy spheres

If q is odd, we are in the same situation we met in the definition of the Kervaire invariant. Now the manifolds we consider have a boundary, but this does not matter. The homomorphism P2q → Z/2 is the Arf invariant defined as before. By construction this homomorphism is injective. It remains to prove that the homomorphisms P2q → Z or Z/2 are onto. We know that elements of P2q can be represented by framed manifolds W2q, which are (q − 1)-connected. Since the boundary bW is a homotopy sphere, by Poincaré duality and the Hurewicz theorem W2q has the homotopy type of a wedge (“bouquet”) of spheres of dimension q. If q ≥ 3, it results from Smale’s h-cobordism theorem that W2q is a q-handlebody. See AppendixD for more on handlebodies. Definition 3.13. A q-handlebody is a manifold W2q of dimension 2q that is obtained by attaching handles of index q to a disc of dimension 2q. In summary, elements of P2q (q ≥ 3) can be represented by parallelizable q- handlebodies such that the intersection form on Hq(W, Z) is unimodular, i.e., of determinant ±1. The unimodularity condition ensures that the boundary bW is a homology sphere. If q , 2 the boundary bW of the handlebody is indeed a homotopy sphere. To prove that the homomorphism P2q → Z or Z/2 is onto, the approach is the following. The technical ingredient is the plumbing construction.

(1) If q is even, the E8-plumbing produces a q-handlebody that is parallelizable with signature 8 and boundary a homotopy sphere. (2) If q is odd, say q = 2k + 1, the argument splits into two subcases. q q (21) If 2q = 4k + 2 is equal to 2, 6, or 14, we consider the product S × S with a suitable normal framing. If we dig a hole in it, i.e., if we consider Sq × Sq \ D˚ 2q, we obtain a framed q-handlebody with Arf invariant 1.

(22) If 2q = 4k + 2 is not equal to 2, 6, or 14 we consider the q-disc bundle associated to the tangent bundle of Sq. Then we plumb two copies of it. We obtain a parallelizable q-handlebody We2q with boundary a homotopy sphere. Its Arf invariant is equal to 1. Note that in this case we do not have a problem analogous to the Kervaire invariant problem, since our manifolds are allowed to have a boundary. The continuation of the story is in next section.

3.5 The Kervaire manifold

In [62], Kervaire constructed a closed topological manifold of dimension 10 which does not have the homotopy type of a diﬀerentiable manifold. We now tell the story. 3.5 The Kervaire manifold 25

Remark. If k is such that the Kervaire invariant KI4k+2 is trivial, then the Arf invariant is equal to 0 for all closed differential stably parallelizable manifolds of dimension (4k + 2), essentially by definition of the Kervaire invariant. Consider the manifold We4k+2 that we have just constructed. Its boundary is homeomorphic to the sphere S4k+1. We glue topologically a (4k + 2)-disc to its boundary to obtain a manifold Q4k+2. By construction, it is a closed topological manifold, equipped with a differential structure outside an open disc. If the differential W4k+2 Q4k+2 structure of e extends over the disc, the so-obtained differentiable manifold diff would be stably parallelizable (it could be framed) and by construction would have Arf invariant 1. It is impossible when the Kervaire invariant KI4k+2 is trivial. Note that the differential structure on We4k+2 extends if and only if the boundary ∂We4k+2 is diffeomorphic to the standard sphere. In his paper [62] Kervaire wanted more (in dimension 10)! So let us forget the differential structure on Q4k+2 outside a disc and let us consider Q4k+2 only as a topological manifold. This topological manifold is the Kervaire manifold in dimension 4k + 2. In [62], Kervaire proceeds as follows. First he shows, nontrivially, that KI10 is trivial. Then he proves that Q4k+2 does not have the homotopy type of a differentiable manifold in any dimension (4k + 2) such that KI4k+2 is trivial. Here are the main steps of the proof, which proceeds by contradiction.

Step 1. Construct an invariant Φ(V) ∈ Z/2 for every triangulable (2k)-connected manifold V of dimension (4k +2) in such a way that Φ(V) = KI4k+2(V) if V is a framed differentiable manifold (it is not required that the differential structure be compatible with the triangulation). The definition of Φ uses cohomology operations. It is a homotopy-type invariant.

Step 2. Prove that Φ(Q4k+2) = 1.

Q4k+2 Step 3. Prove that if there exists a differentiable manifold diff homotopy equiv- Q4k+2 Q4k+2 k alent to , this manifold diff can be framed. If 2 is not congruent to 0 mod 8 the obstruction to framing vanishes by Bott. But Milnor gave a general proof of step 3 in an appendix to Brown–Peterson’s paper [20].

Step 4. Now, KI4k+2(Qdiﬀ) = 1 from steps 1 and 2. It gives the required contradiction each time that KI4k+2 is trivial.

Consequence of the Hill–Hopkins–Ravenel result. The Kervaire manifold Q4k+2 does not have the homotopy type of a diﬀerentiable manifold for 4k +2 , 30, 62, 126. It does admit a diﬀerential structure for 4k + 2 = 30 and 62. The case 4k + 2 = 126 is still open. 26 3 The Kervaire–Milnor study of homotopy spheres

Historical note. When Kervaire wrote his paper, Smale’s (or Stallings’) result on the Poincaré conjecture in higher dimensions was not known. To know that the boundary bWe10 of his plumbing is homeomorphic to the sphere S9 Kervaire relied on a lemma due to Milnor that uses crucially that the plumbing has only two factors. At the same time (or even before), Milnor knew about his E8-plumbing; see [89]. Once Smale’s result was known it was easy to construct (by attaching topologically a disc to the boundary of the E8-plumbing) PL manifolds of dimension congruent to 12 mod 16 that do not have the homotopy type of a diﬀerentiable manifold. The contradiction is obtained from Hirzebruch’s signature formula instead of the Kervaire invariant. The argument is much simpler. Diﬀerential topology was moving fast around the year 1960! 4

Diﬀerentiable knots in codimension ≥3

4.1 On the isotopy of knots and links in any codimension

Definition 4.1. A (spherical) n-knot Kn in codimension q is an n-homotopy sphere n+q n n differentiably embedded in the sphere S . Two n-knots K0 and K1 are equivalent if there exists an orientation-preserving diffeomorphism f : Sn+q → Sn+q such that ( n) n f K0 = K1 (respecting orientations; recall that homotopy spheres are oriented). Comments. (1) As André Haefliger told us, once homotopy spheres had been identified it seemed natural to investigate how they embed in Euclidean space (or equivalently in a sphere). Observe that the theory of knots in high codimension (a little more than the dimension of the sphere) is isomorphic to the “abstract” theory.

(2) For people acquainted with classical knot theory, it might seem better to consider only (images of) embeddings of the standard sphere Sn. But in fact this restriction complicates matters. Images of embeddings of the standard sphere constitute a subset (in fact a subgroup if q ≥ 3) of n-knots.

(3) Beware that an orientation-preserving diﬀeomorphism of Sn+q is usually not isotopic to the identity. However the following proposition is true.

N Proposition 4.2. Let Q0 and Q1 be two compact subsets of S . Suppose that there exists an orientation-preserving diﬀeomorphism Φ : SN → SN such that N N Φ(Q0) = Q1. Then there exists a diﬀeomorphism Ψ : S → S isotopic to the identity such that Ψ(Q0) = Q1.

Comments. (1) As a consequence, knots K0 and K1, which are equivalent by a diffeomorphism preserving the orientation of the ambient sphere, are isotopic. This is also true for links, whatever the codimension may be. Indeed, if Ψt is the ambient isotopy that connects Ψ0 to Ψ1 then Ψt (K0) is an ambient isotopy that connects K0 to K1. (2) If N ≤ 3 it is not necessary to replace Φ by Ψ since in this case any orientation- preserving diffeomorphism of SN is isotopic to the identity. This is nontrivial and was proved by Smale and also by Munkres for N = 1, 2 and by Cerf for N = 3. 28 4 Differentiable knots in codimension ≥3

The proof of the proposition relies on the next lemma. For a proof see [98, Lemma 1.3, p. 523]. Lemma 4.3. Let f : SN → SN be a diffeomorphism preserving the orientation. Let BN ⊂ SN be a differential ball. Then f is isotopic to a diffeomorphism f ∗ which is the identity on BN . To prove the lemma one uses the following result, due to Jean Cerf [25]. Theorem 4.4. Let M N be a connected and oriented differential N-dimensional manifold. It does not need to be compact and may have a boundary. Let f0 and f1 be two orientation-preserving embeddings of the ball BN in Int(M). Then there exists a N N diffeomorphism ϕ : M → M , isotopic to the identity, such that ϕ ◦ f0 = f1. This result is also the key tool to prove that the connected sum of oriented differentiable manifolds is well defined up to orientation-preserving diffeomorphism. For a proof see, e.g., [49, near p. 117]. N Proof of the proposition. If Q1 = S there is nothing to prove. Hence we may N N N N suppose that Q1 , S and let B be a differentiable ball such that Q1 ⊂ B ⊂ S . We apply the lemma with f = Φ−1 and BN as above. Let f ∗ be a diffeomorphism as in the lemma. ∗ ∗ If x ∈ Q0 then Φ(x) ∈ Q1 and hence f (Φ(x)) = Φ(x). Hence f ◦ Φ also sends −1 Q0 to Q1. On the other hand let ft be an isotopy which connects f0 = f = Φ to ∗ −1 ∗ f1 = f . Let ht = ft ◦Φ. We have h0 = f0◦Φ = Φ ◦Φ = id and h1 = f1◦Φ = f ◦Φ. Hence Ψ = f ∗ ◦ Φ is the diffeomorphism we are looking for.

4.2 Embeddings and isotopies in the stable and metastable ranges

The paper that initiated the theory of differentiable knots in codimension ≥3 is André Haefliger’s paper [43]. Here is a short history of the events. In 1936 and 1944 Hassler Whitney published (see [134, 135]) two foundational papers about differentiable manifolds. His results can be summarized as follows. Theorem 4.5 (Embeddings and isotopies in the stable range). Let Mn be a differential n-dimensional manifold. Then, (1) Mn can be differentiably embedded in R2n; (2) any two differentiable embeddings of Mn in R2n+2 are isotopic; (3) if n ≥ 2 and if Mn is connected, any two differentiable embeddings of Mn in R2n+1 are isotopic. 4.3 Below the metastable range 29

Around 1960, Haefliger presented a vast theory of embeddings of differentiable manifolds in differentiable manifolds, announced in [41]. As far as homotopy spheres are concerned his results are as follows. Theorem 4.6 (Embeddings and isotopies in the metastable range). Let Σn be a homotopy n-sphere. Then, (1) Σn can be differentiably embedded in RN if 2N ≥ 3(n + 1); (2) Any two differentiable embeddings of Σn in RN are isotopic if 2N > 3(n + 1).

4.3 Below the metastable range

At about the same time (1960) Christopher Zeeman announced in [136] the following result, proved in [137] (from Smale PL theory it results that PL homotopy spheres of dimension ≥5 are PL isomorphic to Sn). Theorem 4.7. Any two piecewise linear (PL) embeddings of the sphere Sn in RN are PL isotopic if N ≥ n + 3. Naturally the question arose of what happens below the metastable range for differentiable embeddings of homotopy spheres. By general position the complement is simply connected if q ≥ 3. From Zeeman’s result, no topological invariant can be obtained from the knot complement, since it is homeomorphic to Sq−1 × Bn+1. For small values of n the first critical values are n = 3 and N = 6. It was a surprise when Haefliger announced that there exist infinitely many isotopy classes of differentiable embeddings of S3 in S6. The full result is the following ([43]), which indicates that there is an important difference between PL and DIFF. Theorem 4.8. Let k be an integer ≥1. Then there exist infinitely many isotopy classes of differentiable embeddings of S4k−1 in S6k. In order to make, later, a parallel with knots in codimension 2, we give a brief idea of Haefliger’s approach. The following key definition is the starting point.

n Definition 4.9. Let Mi for i = 0, 1 be two oriented closed manifolds differentiably embedded in Sn+q. They are said to be h-cobordant if there exists an oriented manifold W n+1 differentiably and properly embedded in Sn+q × [0, 1] such that

n+1 n+q n (1) W ∩ S × {i} = Mi for i = 0, 1; n+1 n n (2) W is an oriented h-cobordism between M0 and M1 . In particular, bW = M1 − M0. 30 4 Diﬀerentiable knots in codimension ≥3

One of Smale’s many results in [117] reads as follows.

Theorem 4.10 (h-cobordism implies isotopy). Suppose that the differentiable closed n n+q n+1 manifolds Mi in S are h-cobordant. Suppose that the h-cobordism W is simply ≥ ≥ n n connected and that n 5 and q 3. Then M0 and M1 are isotopic. There are two steps in Smale’s proof. In the first step, simple connectivity and n ≥ 5 are used to deduce (h-cobordism theorem) that W n+1 is a product. In the second step, the hypothesis q ≥ 3 is used to straighten the product W n+1 inside Sn+q × [0, 1]. The hypothesis q ≥ 3 is crucial here as we shall see later in the chapter dedicated to “knot cobordism in codimension 2”. For q ≥ 3, Haefliger proves that isotopy classes of embedded n-dimensional homotopy spheres in Sn+q form an abelian group Θn+q,n under the connected sum operation. This is noteworthy, since knots in codimension 2 do not form a group but only a monoid. The inverse is lacking. Haefliger denotes by Σn+q,q the subgroup that consists of embeddings of the standard sphere Sn. The main result of [43] is that Σ6k,4k−1 is isomorphic to the integers Z. The isomorphism is constructed as follows. Suppose that K is an embedded S4k−1 in S6k. Consider S6k as the boundary of the disc B6k+1. Using results of Kervaire, Haefliger proves that K bounds a framed submanifold V4k in B6k+1. Very roughly speaking, there is an obstruction for transforming V4k by surgeries in order to obtain an embedded disc bounding K in B6k+1. This obstruction is an integer that provides the isomorphism between Σ6k,4k−1 and Z. The generator is Haefliger’s famous construction based on a high-dimensional version of the Borromean rings. Once [43] was published, the problem arose of determining the groups Θn+q,n and Σn+q,n in general. The answer was provided by Jerome Levine in [78]. To understand Levine’s results it is necessary to go back a little bit in time. Kervaire and Milnor realized that “their” short exact sequence is in fact a consequence of a large diagram of groups that consists of four braided long exact sequences. This was very likely the intended content of Part II of “Groups of Homotopy Spheres”, which never appeared. Levine proved that the groups Θn+q,n are also parts of a diagram of four braided long exact sequences. His diagram is the unstable version of Kervaire–Milnor’s braided diagram. See also [44]. Levine also describes how to recapture Σn+q,n as a subgroup of Θn+q,n. A consequence of Levine’s result is that the abelian groups Θn+q,n are finitely generated and of rank ≤1 over Q if q ≥ 3. Thus we can say that below the metastable range there exist differentiably knotted homotopy spheres but that, if q ≥ 3, there are rather few of them. This contrasts sharply with the situation q = 2 as we shall see. Comment. André Haefliger has shown us letters that Jerome Levine wrote to him in 1962 and 1963. They indicate that, after 1962, Levine worked very actively on spherical knots in codimensions q ≥ 2. Here is an example he described in a letter 4.3 Below the metastable range 31 dated 7 May 1963 and that was apparently never published. We present this example since it ties together several concepts that are put forward in our book. Let 2k ≥ 4. Let V4k be a parallelizable 2k-handlebody with boundary bV diffeomorphic to S4k−1. Suppose that the signature of V is nonzero. Remark that this signature is a multiple of 8 × order(bP4k) and hence is quite large. Since V4k is a parallelizable handlebody, it is not hard to embed it in S4k+1. In fact it embeds in many ways, the key notion here being the Seifert matrix (see Chapter7). Let φ : V4k → S4k+1 be such an embedding. The spherical knot K that Levine 4k+1 4k+1 N considers is φ(bV) ⊂ S . Let jN : S ,→ S be the standard embedding for any N ≥ 4k + 1.

N Proposition 4.11. The embedding jN φ(bV) ⊂ S is differentiably knotted for any N such that 4k + 1 ≤ N ≤ 6k − 1. Proof. It is enough to prove the statement for N = 6k − 1. 6k−1 6k 6k 6k Consider S as the equator of S . It bounds two balls B+ and B− . Keeping 6k−1 4k 6k K = bV fixed in S we can push the interior of V into the interior of B+ (we neglect jN φ). From now on we argue by contradiction. Suppose that K is differentiably unknotted in S6k−1. Hence it bounds a differentiable disc B4k in S6k−1. We push the interior 4k 6k 4k 4k of B into the interior of B− . The union V ∪ B is a differentiable manifold Vˆ of dimension 4k differentiably embedded in S6k. It is almost parallelizable with nonzero signature. This is impossible. Here is why. 4k Let pk ∈ H (Vˆ ; Z) be the kth Pontrjagin class of Vˆ . Since Vˆ is almost parallelizable, by Hirzebruch’s signature formula the signature of Vˆ is detected by pk. Hence pk is nonzero. By Whitney duality p¯k = −pk where p¯k is the kth Pontrjagin class of the normal bundle ν of Vˆ in S6k. The next lemma provides the contradiction. Lemma 4.12. Let M4k ⊂ S6k be a closed, connected and oriented differential 4k 4k submanifold. Then the dual Pontrjagin class p¯k ∈ H (M ; Z) vanishes. Proof. The normal bundle ν of M4k in S6k is of rank 2k. Let e¯ ∈ H2k(M4k; Z) be the Euler class of the normal bundle ν. From Theorem 31 of Milnor–Stasheff’s lecture 2 notes [95] applied to the normal bundle ν, we have p¯k = e¯ . But e¯ vanishes since it is the normal Euler class of M4k embedded in S6k, by Thom’s definition of the Euler class. See [95, Theorem 14].

The fundamental group of a knot complement

From now on, an n-knot will mean a spherical n-knot in codimension q = 2. A 1-knot is often called a classical knot.A standard n-knot is the image of an embedding of the standard n-sphere. There is a question to be addressed by mathematicians (topologists): How can we represent (or construct) n-knots, for n ≥ 2? There is a diﬃculty there and very likely it prevented the advancement of knots in higher dimensions before the 1960s. At that time two constructions became available: surgery and embeddings (in Sn+2) of parallelizable (n + 1)-manifolds that have a homotopy sphere as boundary (Seifert hypersurfaces). The paper [69] contains a survey of the theory of n-knots before 1960.

5.1 Homotopy n-spheres embedded in Sn+2

Proposition 5.1. Let Ln be a closed oriented manifold without boundary embedded in Sn+2. Then its normal bundle is trivial.

Proof. Let us observe ﬁrst that the normal bundle is orientable. Since it has rank 2, it is enough to prove that it has a nowhere-vanishing section. The only obstruction to construct such a section is the Euler class e ∈ H2(Ln; Z). Since H2(Sn+2; Z) = 0, Thom’s formula for the Euler class implies that e = 0. Observe that the same proof works if Sn+2 is replaced by an orientable manifold W n+2 such that H2(W n+2; Z) = 0.

Construction of knots by surgery. Kervaire uses this result in the following way to construct several n-knots by surgery. Suppose that the n-knot K is the standard sphere Sn embedded in Sn+2. Thanks to the proposition above we can perform surgery on K. Moreover, if n ≥ 2 there is a unique trivialization of the normal bundle. Therefore, there is a well-defined manifold Mn+2 obtained by doing surgery on the knot K ⊂ Sn+2. Kervaire’s idea is to construct this manifold Mn+2 first and then perform the surgery backwards to get the knot. Observe that this procedure works only for knots represented by an embedding of the standard sphere. Then let K × B2 ⊂ Sn+2. Let E be the knot exterior defined by Sn+2 \(K × B˚2). 34 5 The fundamental group of a knot complement

The advantage of the knot exterior over the knot complement Sn+2 \ K is that it is a compact manifold with boundary, having the same homotopy type as the complement.

Lovely observation. The exterior E is the common part of the surgery. Denote by Mn+2 the manifold obtained by doing surgery on K in Sn+2. Remark that, by construction, Mn+2 contains the scar (an embedded circle) γ ⊂ Mn+2 such that if we perform a surgery on γ we get the manifold Sn+2 with its knot K inside as the scar. The next proposition is an easy consequence of Alexander and Poincaré dualities. Proposition 5.2. Let n ≥ 2 and let Mn+2 be the manifold obtained by surgery on a standard n-knot in Sn+2. Then Mn+2 has the homology of S1 × Sn+1. Explicitly, ( n+2 Z if i = 0, 1, n + 1, n + 2, Hi(M ; Z) = 0 otherwise.

Later we shall see several applications of Kervaire’s construction of a manifold like Mn+2. More information on the knot exterior is given in Section 6.1.

Seifert hypersurfaces. The next theorem was proved by Kervaire in [63], but it was certainly known to most differential topologists at the time of the meeting in the honor of Marston Morse. Theorem 5.3 (Kervaire). Let Σn be a homotopy n-sphere differentiably embedded in Sn+2. Then there exists an orientable (n + 1)-differentiable manifold Fn+1 ⊂ Sn+2 such that bFn+1 = Σn. Definition 5.4. A submanifold such as Fn+1 is called a Seifert hypersurface of the knot Σn. Corollary 5.5. A homotopy n-sphere can be embedded in Sn+2 if and only if it bounds a parallelizable manifold. In other words, the elements of Θn that can be differentiably embedded in Sn+2 are exactly the elements of bPn+1. The existence of Seifert hypersurfaces is proved in a more general context in AppendixB. The corollary is discussed in Proposition E.2.

Remark on the groups Θn+q,n for q ≥ 3. We have just stated that every homotopy n-sphere embedded in Sn+2 has a trivial normal bundle and bounds a framed (n + 1)- dimensional submanifold in Sn+2. Neither of these statements holds for homotopy spheres embedded in codimension ≥3. More precisely, Levine in [78] constructs two homomorphisms: 5.2 Necessary conditions to be the fundamental group of a knot complement 35

n+q,n (1) a homomorphism Θ → πn−1(SOq) expressing that the obstruction of the normal bundle of Σn ,→ Sn+q is trivial;

n+q,n (2) a homomorphism Θ → πn(Gq, SOq) that expresses the obstruction of the embedded Σn ,→ Sn+q bounding a framed submanifold in Sn+q. See Levine’s paper [78] for a deﬁnition of the H-space Gq.

These two homomorphisms take place in two long exact sequences that are the essential constituent of Levine’s diagram of four braided long exact sequences (the unstable version of Kervaire–Milnor’s).

5.2 Necessary conditions for a group to be the fundamental group of a knot complement

Kervaire’s paper that got knots in higher dimensions really started was his determination of the fundamental group.

Theorem 5.6 (Kervaire [63]). Let K be an n-knot (n ≥ 1). Let Γ be the fundamental group of the complement Sn+2 \ K. Then Γ satisﬁes the following conditions:

(1) H1(Γ) = Z;

(2) H2(Γ) = 0; (3) Γ is ﬁnitely presented;

(4) Γ is the normal closure of a single element.

Comments. (1) The reason why we state the Kervaire conditions in this order will be clear in what follows.

(2) Usually the weight of a group is deﬁned to be the minimal number of elements of the group such that the normal closure of this set of elements is the whole group. Condition (4) can hence be restated: Γ is of weight 1. Another way to state this condition is that there exists an element z ∈ Γ such that every element of Γ is a product of conjugates of z and z−1.

(3) The notation Hi(Γ) denotes the ith homology group of Γ with integer coeﬃcients Z and with the trivial action of Γ on Z.

0 (4) We recall that H1(Γ) = Γ/Γ is the abelianization of Γ.

(5) To understand H2(Γ), the key result is Hopf’s theorem on H2(Γ). 36 5 The fundamental group of a knot complement

Theorem 5.7 (Hopf’s theorem). Let X be a connected “good” topological space (e.g., a C.W. complex) with fundamental group Γ. Let h2 : π2(X) → H2(X; Z) be the Hurewicz homomorphism. Then the quotient H2(X; Z)/Im(h2) depends only on Γ and is by deﬁnition H2(Γ). Corollary 5.8. The following statements are equivalent:

(1) H2(Γ) = 0.

(2) For all spaces X as in the theorem, the Hurewicz homomorphism h2 : π2(X) → H2(X; Z) is onto.

(3) There exists a space X such that H2(X; Z) = 0. Proof of Theorem 5.6. Conditions (1) and (2) are an immediate consequence of Alexander duality. Indeed, let Z ⊂ Sn+2 be a compact subset of Sn+2 that has the Cech˘ cohomology of Sn. Then Sn+2 \ Z has the singular homology of S1. In particular, conditions (1) and (2) are also true for wild embeddings of Sn in Sn+2. For conditions (3) and (4) to be satisfied requires more on the embedding. To simplify matters let us assume that K is a homotopy n-sphere topologically embedded in Sn+2 with a neighborhood homeomorphic to K × B2. Let E be the exterior Sn+2 \(K × B˚2). It is clear that E has the same homotopy type as the knot complement Sn+2 \ K. Since E is a compact topological manifold, its fundamental group is finitely presented. This proves that condition (3) is satisfied. We can reconstruct Sn+2 from the exterior by attaching to E a 2-cell e2 = {x}× B2 n+2 for some x ∈ K and then an (n + 2)-cell e . Let z ∈ π1(E) be represented by the loop {x} × ∂B2. Since n ≥ 1 and since Sn+2 is simply connected, E ∪ e2 is simply connected. Hence the normal subgroup of π1(E) generated by z is equal to the whole 2 group. This proves condition (4). The loop {x} × ∂B and its class z ∈ π1(E) are often called a meridian.

A comment about meridians and killers. The following terminology seems to be established. A killer γ is an element of a group Γ such that its normal closure is equal to the whole group. Define two killers γ and γ∗ to be equivalent if there exists an automorphism Ψ : Γ → Γ such that Ψ(γ) = γ∗. It is quite natural to ask whether there exist knot groups with killers nonequivalent to a meridian. The answer is yes. In [114] Silver–Whitten–Williams prove that there exist classical knot groups with infinitely many nonequivalent killers. For instance, this is the case for rational knots and for torus knots. The authors put forward the exciting (or risky?) conjecture that every classical knot group (of course nonabelian) has infinitely many nonequivalent killers. Jonathan Hillman in [48] calls a killer a weight element. He introduces the following convenient terminology: A weight orbit is the equivalence class of a 5.3 Sufficiency of the conditions if n ≥ 3 37 weight element through group automorphisms; a weight class is the equivalence class through conjugation. He proves that knot groups that are metabelian have only one weight class. Classical knot groups are not metabelian by Neuwirth [100].

5.3 Suﬃciency of the conditions if n ≥ 3

We begin this section by presenting Kervaire’s construction of knots by surgery. Deﬁnition 5.9. Let n ≥ 2.A knot manifold Mn+2 is a closed, connected and oriented diﬀerentiable manifold of dimension (n + 2) such that n+2 1 n+1 (1) H∗(M ; Z) = H∗(S × S ; Z);

(2) π1(M) has weight 1. Surgery on a spherical n-knot in Sn+2 produces a manifold satisfying condition (1) (see Proposition 5.2) together with a killer (see the proof of Theorem 5.6). n+2 Conversely, let M be a knot manifold and z ∈ π1(M) a weight element. Following Kervaire, we now show how to obtain a spherical n-knot from this data by surgery if n ≥ 3. Let z ∈ π1(M) be the chosen killer. We represent this element by an embedded circle ζ : S1 ,→ M. Note that the isotopy class of this embedding is well defined, once z is chosen (only its weight class matters). Claim. The normal bundle of ζ : S1 ,→ M is trivial and there are two homotopy classes of trivializations of this bundle. The first assertion is true since M is orientable and the second one is true since homotopy classes of trivializations are in bijection with π1(SOn+1). We choose a trivialization S1 × Bn+1 ,→ M and perform a surgery on S1 × Bn+1. Let M 0 be the manifold obtained by the surgery. Proposition 5.10. The manifold M 0 is a homotopy sphere. Proof. Let V = M \(S1 × B˚n+1) be the common part of M and M 0. The manifold M is obtained from V by attaching an (n+1)-cell and an (n+2)-cell. Hence V and M have the same n-skeleton. Since n ≥ 3 we have π1(V) = π1(M) = Γ and Hi(V; Z) = Hi(M; Z) = 0 for 2 ≤ i ≤ n − 1. On the other hand, M 0 is obtained from V by attaching a 2-cell and an (n +2)-cell. By construction the 2-cell “kills” the element z. Since z is of infinite order in H1(V; Z) the 2-cell does not create torsion in H2. Since z is a killer, the fundamental group 0 0 π1(M ) is trivial. Moreover, Hi(M ; Z) = 0 for 1 ≤ i ≤ n − 1. Hence, by Poincaré duality, M 0 is a homology sphere and, since its fundamental group is trivial, M 0 is a homotopy sphere. 38 5 The fundamental group of a knot complement

The homotopy sphere M 0 has some differential structure but we cannot guarantee that this structure is the standard one. Since its dimension (n + 2) is ≥5 we can perform a connected sum with its inverse in the group Θn+2 to obtain the standard differential structure. This connected sum can take place on a small (n + 2)-ball. We 0 0 n+2 write Mmod for this modified differential structure such that Mmod = S . 0 n+2 0 Where is the knot K in Mmod = S ? Well Mmod is constructed by surgery. The 0 common part is the knot exterior, and the knot is the scar of the surgery in Mmod. Let us remark for future reference that the knot exterior has the same n-skeleton as M. Remark. One defect of the construction of knots by surgery is that it produces only standard knots. This is the price we pay to work in the differential category. It is possible to mimic this construction in the PL or in the TOP category, once adequate hypotheses are formulated to ensure the existence of trivial normal bundles. See Kervaire–Vasquez [68] for the PL case and Hillman [48] for the TOP case.

Gluck’s reconstruction. We have noticed in the claim above that the normal bundle of the embedding ζ : S1 → Mn+2 has two trivializations. Hence we can construct a priori two n-knots by surgery. By deﬁnition, these two knots diﬀer by Gluck’s reconstruction. Clearly they have the same exterior. We shall see in Chapter6 that there are at most two knots with the same exterior.

4 The case n = 2. Let M be a knot manifold and z ∈ π1(M) be a weight element. We perform a surgery on z. We get a homotopy 4-sphere Σ4 together with a 2-knot in it. Today it is not known whether we can modify the differential structure on Σ4 to get a 2-knot in S4. Fifty years ago this was presumably seen as a shortcoming. Today it is quite the contrary. Maybe this is the way to construct counterexamples to the differential Poincaré conjecture in dimension 4! One difficulty (among several) is constructing 4-dimensional knot manifolds. Examples are due to Cappell and Shaneson. See [24]. Kervaire’s realization theorem is the following. Theorem 5.11 (Kervaire). Let Γ be a group satisfying conditions (1) to (4) of The- orem 5.6. Suppose that n ≥ 3. Then there exists an embedding of the standard n-sphere K ,→ Sn+2 such that the fundamental group of the exterior is isomorphic to Γ. The first step in the proof of the realization theorem is the next proposition. Proposition 5.12. Let Γ be a group satisfying conditions (1), (2), and (3) of the theorem. Let n ≥ 3. Then there exists a differentiable manifold Mn+2, orientable, compact, connected, without boundary such that

(i) π1(M) = Γ; 5.3 Suﬃciency of the conditions if n ≥ 3 39

(ii) Hi(M; Z) = Z if i = 0, 1, n+1, n+2 and =0 otherwise (in other words, additively 1 n+1 H∗(M; Z) = H∗(S × S ; Z)).

Clearly, the theorem follows from a surgery on the knot manifold provided by the proposition applied to a group satisfying conditions (1) to (4). Before the proof of Proposition 5.12 we state and prove another proposition.

Proposition 5.13. Let Π be group satisfying conditions (2) and (3). Then there exists a compact and connected C.W. complex Y of dimension 3 such that

(i) π1(Y) = Π;

(ii) Hi(Y; Z) = 0 for i ≥ 2.

Proof. We present the proof with some details in order to show how condition (2) and Hopf’s theorem are used. (Conditions (1) and (4) are not necessary.) Choose arbitrarily a finite presentation of the group Π. Then construct a fi- 0 0 nite C.W. complex Y of dimension 2 with π1(Y ) = Π, following the instructions contained in the following presentation: one 0-cell, the 1-cells correspond to the generators, and the 2-cells correspond to the relations. We then consider the chain complex of Y 0, with integer coefficients,

0 ∂2 0 ∂1 0 C2(Y ) −→ C1(Y ) −→ C0(Y ).

0 0 0 We have H2(Y ; Z) = Ker(∂2) since C3(Y ) = 0. Hence H2(Y ; Z) is a ﬁnitely 0 generated free abelian group, of rank say s. Let {g1,..., gs } be a basis of H2(Y ; Z). 0 By construction π1(Y ) = Π and since H2(Π) = 0, by Hopf’s theorem, the Hurewicz 0 0 homomorphism h2 : π2(Y ) → H2(Y ; Z) is onto. Represent then each gi by a map 2 → 0 3 3 0 3 ϕi : S Y . We attach s 3-cells e1,..., es on Y , where ei is attached by the map ϕi for i = 1,..., s. We denote by Y the 3-dimensional C.W. complex thus obtained. We claim that this is the complex we are looking for.

0 (i) Since Y is obtained by attaching cells of dimension 3 to Y we have π1(Y) = 0 π1(Y ) = Π.

(ii) Consider the following portion of the chain complex of Y:

∂3 ∂2 C3(Y) −→ C2(Y) −→ C1(Y).

By construction we have Ker ∂2 = Im ∂3 and hence H2(Y; Z) = 0. Now H3(Y; Z) = Ker(∂3) since there are no 4-cells in Y. But Ker(∂3) = 0 since the basis elements of C3(Y) are sent by ∂3 to linearly independent elements. 40 5 The fundamental group of a knot complement

Proof of Proposition 5.12. Let Γ be a group satisfying the Kervaire conditions (1) to (4). Let Y be a C.W. complex supplied by Proposition 5.13 such that π1(Y) = Γ. The idea for constructing the manifold Mn+2 is to first construct a manifold N n+3 that has the complex Y for a spine. The manifold Mn+2 is then the boundary ∂N n+3. The fact that ∂N n+3 has the required properties is easy. Since the codimension of Y in N n+3 is n ≥ 3, we have π1(M) = π1(N) = π1(Y). Poincaré duality implies that the homology of M is as desired. There are at least two (closely related indeed) ways to construct N n+3. The first method is to attach handles to a (n+3)-dimensional disc. The handles are of indexes 1, 2, and 3 and attached step by step in a way similar to the way the complex Y is constructed. This is the path followed by Kervaire in [63, pp. 110–111]. In fact Kervaire uses the surgery technique (certainly not a surprise). This is equivalent to construct a thickening of Y. Kervaire’s argument works for n ≥ 3. A more expeditious method is to construct a finite 3-dimensional simplicial complex Y ∗ homotopy equivalent to Y. We then embed Y ∗ piecewise linearly in Rn+3. The manifold N n+3 is a regular neighborhood of Y ∗ ,→ Rn+3. Since the simplicial complex Y ∗ is of dimension 3, by general position it can be embedded in n+3 ∗ 6 R for n ≥ 4. But, since H3(Y ) = 0 it can also be embedded in R . Hence both methods produce knots in Sn+2 for n ≥ 3. Remark. The proof given above for Theorem 5.11 shows that any group satisfying the Kervaire conditions can be realized as the fundamental group of the exterior of an embedding of the standard n-sphere. We have seen above that a homotopy n-sphere can be embedded in Sn+2 if and only if it bounds a parallelizable manifold. In [63] Kervaire proves, following an argument of Milnor’s, that any group satisfying the Kervaire conditions can be realized as the fundamental group of the exterior of an embedding of any given element of bPn+1.

5.4 The Kervaire conjecture

Question. Can a free product G ∗ Z be a group satisfying the Kervaire conditions?

More precisely, we suppose that G is finitely presented and satisfies H1(G) = 0 = H2(G). Then clearly G ∗ Z satisfies the Kervaire conditions (1), (2), and (3). Is it possible that this free product also satisfies condition (4)? In [63] Kervaire studied the free product I120 ∗ Z where I120 is the binary icosahedral group of order 120. He proved that I120 ∗ Z is not of weight 1. See [63, p. 117]. Incidentally this proves that condition (4) is not a consequence of conditions (1), (2), and (3). Kervaire asked whether it is possible that a nontrivial free product G ∗ Z can ever be of weight 1. The negative answer to this question is known as the Kervaire conjecture, although 5.5 Groups that satisfy the Kervaire conditions 41

Kervaire told us that he asked the question (in particular to Gilbert Baumslag) but did not formulate the conjecture. It is known that the Kervaire conjecture is true for torsion-free groups. See Klyachko [70] and Fenn–Rourke [35]. Remark. Let G be a group satisfying the following three conditions:

(i) H1(G) = 0;

(ii) H2(G) = 0; (iii) G is ﬁnitely presented. In [65] Kervaire proved that these are exactly the groups that are the fundamental group of a homology sphere of dimension ≥5. His arguments are reminiscent of those about knot groups.

5.5 Groups that satisfy the Kervaire conditions

Theorem 5.6 says that the Kervaire conditions are necessary for all n ≥ 1 and Theorem 5.11 says that they are suﬃcient if n ≥ 3. Question. Can we improve on Theorem 5.11? We shall see that the answer is no. It is easy to deduce from Artin’s spinning construction (1925) that there are inclusions of the following sets of groups up to isomorphism: Groups of Groups of Groups of Groups of ⊂ ⊂ = . 1-knots 2-knots 3-knots n-knots for n ≥ 3 It is known that each inclusion is strict (see below). Among the set of groups (up to isomorphism) that satisfy the Kervaire conditions the subset of groups of classical knots is very small. The 2-knot groups constitute an interesting intermediate class. Hillman’s book [48] contains a lot of information about them. Let Γ be an n-knot group (for any n) and let Γ0 be its commutator subgroup. Of course, condition (1) says that Γ/Γ0 = Z. Implicitly we often assume that Γ , Z. Questions. Can Γ0 be abelian? If the answer is positive, what abelian groups can be isomorphic to Γ0? For n = 1, the answer is no. Indeed, from Neuwirth’s analysis of the commutator subgroup of a 1-knot group [100], the group Γ0 contains a free subgroup of rank ≥2 (the fundamental group of an incompressible Seifert surface). The case n = 1 is in sharp contrast with the case n ≥ 3, since Hausmann–Kervaire prove in [45] the following result. 42 5 The fundamental group of a knot complement

Theorem 5.14. Let G be a ﬁnitely generated abelian group. Then there exists an n-knot group Γ for n ≥ 3 such that Γ0 is isomorphic to G if and only if G satisﬁes the following conditions:

(1) rG , 1, 2; k (2) rG(2 ) , 1, 2 for every k ≥ 1; k (3) rG(3 ) is equal to 1 for at most one value of k ≥ 1. Definition 5.15. For any abelian group G (not necessarily finitely generated) the rank rG is the dimension of the Q-vector space G ⊗Z Q. For a prime p and an integer k ≥ 1, the number of factors isomorphic to Z/pk in the torsion subgroup of G is k denoted by rG(p ). Hausmann and Kervaire also prove that an abelian Γ0 can be nonfinitely generated and give some examples. It seems that a complete classification of nonfinitely generated abelian Γ0 is not known. The case n = 2 is somewhat in between. See [83] and [48]. Theorem 5.16. If the commutator subgroup Γ0 of a 2-knot group Γ is abelian, then Γ0 3 [ 1 ] is isomorphic to one of the following groups: Z , Z 2 , or a finite cyclic group of odd order. Question. What is the center Z(Γ) of a knot group Γ? The answer for the classical case n = 1 is the following: (1) If Z(Γ) is nontrivial it is isomorphic to the integers Z. This result is due to Neuwirth [100]. (2) Burde–Zieschang [22] proved that Z(Γ) = Z if and only if the knot is a torus knot. Again the situation is very different if n ≥ 3. The next result is also due to Hausmann– Kervaire. See [46]. Theorem 5.17. Let G be a finitely generated abelian group. Then there exists a knot group Γ for n ≥ 3 such that Z(Γ) is isomorphic to G. It seems that it is unknown whether the center can be nonfinitely generated. The case n = 2 is again in between. For more details see [48]. Theorem 5.18. The center Z(Γ) of a 2-knot group is of rank ≤2. If the rank is 2, then Z(Γ) is torsion-free. Hillman gives examples of 2-knot groups Γ with Z(Γ) isomorphic to Z2, Z, Z ⊕ Z/2, Z/2. 6

Knot modules

6.1 The knot exterior

Let us recall J. H. C. Whitehead’s theorem. Deﬁnition 6.1. Let X and Y be two connected C.W. complexes with basepoint. A basepoint-preserving map f : X → Y is a homotopy equivalence if there exists a basepoint-preserving map g : Y → X such that the two composition maps f ◦ g and g ◦ f are homotopic to the identity, keeping the basepoints ﬁxed. Theorem 6.2. Let X and Y be two connected C.W. complexes with basepoint. Let f : X → Y be a basepoint-preserving map. The following three statements are equivalent: (1) f is a homotopy equivalence;

(2) f induces isomorphisms fj : πj(X) → πj(Y) for all integers j ≥ 1;

(3) f1 : π1(X) → π1(Y) is an isomorphism and efj : Hj(Xe; Z) → Hj(Ye; Z) is an isomorphism for every j ≥ 2 where ef : Xe → Ye is a lifting of f between universal coverings. Knot invariants are provided mostly by the knot exterior E(K) (and its coverings). Since E(K) has the homology of a circle, the homology of E(K) is useless. Well, let us look at the homotopy groups πj(E(K)) for j ≥ 2. Let us recall that a connected C.W. complex X is said to be aspherical if πj(X) = 0 for j ≥ 2. Note that nothing is required on the fundamental group (of course!). By Whitehead’s theorem, X is aspherical if and only if its universal covering is contractible. Question. Can a knot exterior be aspherical? The answer is very contrasting. By Papakyriakopoulos [102] the knot exterior of a 1-knot is always aspherical. But, for n ≥ 2, if the exterior of an n-knot is aspherical then the knot group is isomorphic to Z. This result is due to Dyer–Vasquez [31]. Hence for n ≥ 2, an n-knot exterior that is aspherical has the homotopy type of a circle. The next theorem was known when Kervaire wrote his Paris thesis. See [77]. Theorem 6.3 (Levine’s unknotting theorem). Let Kn ⊂ Sn+2 be an n-knot such that E(K) has the homotopy type of a circle. Assume that n ≥ 3. Then K is the trivial knot. 44 6 Knot modules

Comments. (1) Levine’s result is true in the differential and also in the PL (locally flat) categories. It was already known that the result is true in the topological (locally flat) category, thanks to Stallings [119]. (2) For n = 1 Levine’s result is true by Dehn’s lemma. For n = 2 the result is true in the TOP category by Freedman. See Hillman’s book [48, pp. 4–5]. Indeed, the hypothesis π1(E(K)) = Z is enough. Let us now come back to the homotopy groups πj(E(K)) for j ≥ 2. Since the fundamental group is never trivial, these groups are not only abelian groups but modules over the group ring Zπ1(E(K)). Except when the fundamental group is infinite cyclic, this group ring is never commutative. It seems that the homotopy groups πj(E(K)) are little studied when the fundamental group is not infinite cyclic. Let us write the infinite cyclic group in a multiplicative way as T = {tk } for k ∈ Z. Hence the group ring ZT of the infinite cyclic group can be identified with the ring of Laurent polynomials Z[t, t−1]. Definition 6.4. Let Eb(K) → E(K) be the infinite cyclic covering of E(K), i.e., the covering associated with the abelianization epimorphism of the fundamental group π1(E(K).A knot module is a homology group Hi(Eb(K); Z) equipped with the ZT-module structure induced by the Galois transformations. In the case of 1-knots, knot modules are often called Alexander modules. In Chapter II of his Paris paper [64], Kervaire investigates the following situation: n+2 1 q ≥ 2 is an integer and K ⊂ S is an n-knot such that πi(E(K)) = πi(S ) for i < q and πq(E(K)) , 0.

Problem. Determine πq(E(K)) = Hq(Eb(K); Z) as a ZT-module. In today’s words Kervaire wishes to understand the “first nontrivial knot module” but he states his results in terms of πq(E(K)). Later he regretted having adopted this viewpoint, since many of his arguments are valid for knot modules in general, not only for the first nontrivial one. Thus he found Jerome Levine’s general study of knot modules in [82] much better. The next lemma contains useful information. It is proved along the way by Levine in [77]. It is also a consequence of Poincaré duality in the infinite cyclic covering. n n+2 1 Lemma 6.5. Let K ⊂ S be an n-knot such that πi(E(K)) = πi(S ) for 1 ≤ i ≤ q = (n + 1)/2. Then E(K) has the homotopy type of S1, and hence the knot is trivial.

Thus the problem of the determination of πq(E(K)) splits into three cases which shall be treated separately below: (1) q < n/2; (2) n is even and q = n/2; (3) n is odd and q = (n + 1)/2. 6.2 Some algebraic properties of knot modules 45

Historical note. Clearly, one question is, how much of an n-knot is detected from its exterior? Here are some answers.

(A) An n-knot is determined by its exterior if n = 1 by Gordon–Luecke [39].

(B) For n ≥ 2 there are at most two n-knots with a diﬀeomorphic exterior. This result is due to Gluck [37] for n = 2, to Browder [17] for n ≥ 5, and to Lashof– Shaneson [73] for n ≥ 3. An equivalent statement is that there are at most two admissible meridians on the boundary bE(K). The proofs of these theorems are stated for embeddings of the standard sphere. In [23] Cappell and Shaneson prove that there do exist pairs of inequivalent n-knots with the same exterior.

(C) Let us now consider the homotopy type of the exterior. There are two ways to proceed: We consider either the exterior E(K) or the pair (E(K), bE(K)). If n = 1 by the asphericity of knots, the homotopy type of E(K) is determined by the fundamental group π1(E(K)) and the homotopy type of the pair (E(K), bE(K)) is determined by the pair of groups (fundamental group, periph- eral subgroup). The work of Waldhausen [126] and Johannson [54] implies then that the homotopy type of the pair (E(K), bE(K)) determines the exterior. If n ≥ 3, Lashof–Shaneson [73] prove that the homotopy type of the pair (E(K), bE(K)) determines the diﬀerential type of the knot exterior, provided that the fundamental group of the exterior is inﬁnite cyclic. Their proof assumes that the n-knot is represented by an embedding of the standard sphere.

6.2 Some algebraic properties of knot modules

Let us recall some easy properties of the commutative ring with unit ZT.

(A) ZT is an integral domain (no zero divisors) but it is not a principal ideal domain, although it might be tempting to think it is since QT is. As a consequence there is no decomposition of finitely generated ZT-modules as a direct sum of cyclic modules. In fact, a classification of finitely generated ZT-modules seems inaccessible.

(B) ZT is a noetherian ring (Hilbert basis theorem). As a consequence, every submodule of a finitely generated ZT-module is also finitely generated and finitely generated modules are finitely presented. A surjective endomorphism of a finitely generated ZT-module is an isomorphism. 46 6 Knot modules

(C) ZT is a unique factorization domain (Gauss lemma). One consequence is that Fitting ideals of a knot module have a gcd. This gives rise to Alexander invariants. We now turn to knot modules and recall a definition due to Levine. Notice that the letter “K” is the first letter of knot as well as Kervaire. Definition 6.6. A ZT-module H is said to be of type K if it is finitely generated and if multiplication by (t − 1) is an isomorphism of H. Theorem 6.7 (Kervaire). Knot modules are of type K. Proof. The knot exterior E(K) can be triangulated as a finite simplicial complex. Hence the chain group Ci(Eb(K); Z) is a finitely generated free ZT-module. Since the ring ZT is noetherian, the cycles are finitely generated as a module and hence the quotient module Hi(Eb(K); Z) is also a finitely generated module. Let us now prove that multiplication by (t − 1) is an isomorphism. The key ingredient used by Kervaire is the spectral sequence of a covering map. Since the Galois group is the integers (written multiplicatively) T, this spectral sequence degenerates at the E2 term and it is better replaced by an exact sequence due originally to Serre (appendix to his thesis) and much used later in knot theory. Lemma 6.8 ((t − 1) lemma). Let X be a connected complex (simplicial or C.W.) and let π1(X) → T be a surjective homomorphism. Let p : Xb → X be the projection of the infinite cyclic covering associated to this homomorphism. Then (t −1) : Hi(Xb) → 1 Hi(Xb) is an isomorphism for all i ≥ 1 if and only if H∗(X) is isomorphic to H∗(S ). Proof. It is easy to see that one has a short exact sequence of chain complexes (with integer coefficients)

t−1 p∗ 0 → C∗(Xb) −→ C∗(Xb) −→ C∗(X) → 0. Its associated homology long exact sequence is the object to be considered:

t−1 pi · · · → Hi+1(X) → Hi(Xb) −→ Hi(Xb) −→ Hi(X) → · · · .

This exact sequence implies that (t − 1) : Hi(Xb) → Hi(Xb) is an isomorphism for all 1 i ≥ 2 if and only if H∗(X) is isomorphic to H∗(S ). The argument also works for i = 1: since (t − 1) : H0(Xb)−→H0(Xb) is the zero homomorphism, the connecting homomorphism δ : H1(X)−→H0(Xb) is a isomorphism and p1 = 0. We apply the lemma to the knot exterior E(K) which has the homology of a circle. An easy consequence of the theorem is that a knot module is a torsion ZT-module. 6.2 Some algebraic properties of knot modules 47

Deﬁnition 6.9. Let H be a ZT-module of type K. We write Tors(H) for the Z-torsion submodule of H and f (H) (following Levine) for the quotient module H/Tors(H). Theorem 6.10 (Kervaire Tors(H)). Let H be a ZT-module of type K. Then the cardinal of Tors(H) is ﬁnite. For a proof see Kervaire [64, Lemme II.8] or Levine [82, Lemma 3.1].

Caution. Note that f (H) is finitely generated as a module. A lot of examples from classical knot theory show that it is not finitely generated as an abelian group, in general. Here are some deeper properties of ZT. (D) ZT is of (global) homological dimension 2. For details see [85, Chapter VII]. This means that the following three equivalent properties are satisfied: k 0 0 (i) ExtZT (H, H ) = 0 for all ZT-modules H and H and all integers k > 2. (ii) For every ZT-module H and for every exact sequence with projective ZT-modules Pi, for i = 0, 1,

0 → C2 → P1 → P0 → H → 0;

the module C2 is also projective. (iii) For every ZT-module H, there exist projective resolutions of length at most 2. (E) ZT satisfies the Serre conjecture: every finitely generated projective ZT-module is free. See [123]. A consequence is the following. Π Proposition 6.11. Let H be a knot module. Let F1 −→ F0 → H → 0 be a finite presentation of H. Then the kernel of Π is a finitely generated free ZT-module. Hence, there exists a resolution 0 → F2 → F1 → F0 → H → 0 of H where the Fi are finitely generated free ZT-modules, for i = 0, 1, 2. Here is a pretty result by Levine. See [82, Proposition 3.5]. Theorem 6.12. Let H be a ZT-module of type K. Then the following statements are equivalent: (1) Tors(H) = 0; (2) H is of homological dimension 1; (3) H possesses a square presentation matrix. 48 6 Knot modules

6.3 The qth knot module when q < n/2

The theorem proved by Kervaire ([64]) is the following.

Theorem 6.13. Let 1 < q < n/2 and let H be a ZT-module. Then there exists an n n+2 n-knot K ⊂ S with Hi(Eb(K); Z) = 0 if 0 < i < q and Hq(Eb(K); Z) = H if and only if H is of type K.

Proof. We already know that type K is necessary. So let us prove that this condition is suﬃcient. The proof shows that the knot can be realized by an embedding of the standard diﬀerential sphere. In the proof we shall need the following statement of the Hurewicz theorem.

Theorem 6.14 (Hurewicz). Let X be a connected C.W. complex such that πi(X) = 0 for 1 ≤ i < k. Then πk(X) → Hk(X, Z) is an isomorphism and πk+1(X) → Hk+1(X, Z) is onto. The proof follows the same steps as the proof of the characterization of the knot group. Proposition 6.15. Let H be a ZT-module of type K. Then there exists a connected ﬁnite C.W. complex Y of dimension ≤(q + 2) such that

(1) π1(Y) = Z;

(2) Hi(Yb; Z) = 0 for 1 < i < q;

(3) Hq(Yb; Z) = H;

(4) Hi(Yb; Z) = 0 for q < i. If we admit the proposition, the proof of the theorem is achieved as follows. The homology exact sequence of the inﬁnite cyclic covering Yb reveals that Y has the homology of a circle. We choose a simplicial complex of dimension (q + 2) that has the same homotopy type as Y and denote it again by Y. It can be piecewise linearly embedded in Rn+3 since its top-dimension homology vanishes. This is an easy consequence of Shapiro [113, Theorem 7.2]. We denote by M the boundary of a regular neighborhood N of the embedded Y in Rn+3. The PL (n + 2)-manifold M can be smoothed by Hirsch–Whitehead. It has the same (q + 1)-type as Y. The proof is achieved by surgery on M as in the characterization of the knot group, since M is a knot manifold.

Proof of Proposition 6.15. We choose a ﬁnite presentation of the ZT-module H: dq+1 Fq+1 −→ Fq → H → 0. We know from the preceding section that the kernel 6.3 The qth knot module when q < n/2 49

Ker(dq+1) is free and ﬁnitely generated. We then have a free resolution

dq+2 dq+1 0 → Fq+2 −→ Fq+1 −→ Fq → H → 0.

We denote by ri the ZT-dimension of Fi for i = q + 2, q + 1, q. We shall construct the C.W. complex Y, skeleton by skeleton, using the chosen resolution of H. We begin with a 0-cell ∗ and attach to it one 1-cell. We thus obtain a circle. To the 0-cell ∗ we attach rq cells of dimension q. We thus obtain a wedge of 1 S with rq spheres of dimension q. We denote this complex by Yq. The inﬁnite cyclic covering pq : Ybq → Yq is also its universal covering. By construction Cq(Ybq; Z) is isomorphic to Fq as a ZT-module. { 1 2 rq+1 } Z ( i ) Let q+1, q+1, . . ., q+1 be a T-basis of Fq+1. Consider the element dq+1 q+1 ∈ Fq = Hq(Ybq; Z) = πq(Ybq) = π(Yq). For each 1 = 1,..., rq+1 we attach a (q + 1)- i ( i ) cell eq+1 to Yq by the element dq+1 q+1 . We thus obtain a C.W. complex Yq+1 of dimension (q + 1). Let pq+1 : Ybq+1 → Yq+1 be the universal covering. Far from dimensions 0 and 1 the chain complex of Ybq+1 is isomorphic over ZT to 0 → Fq+1 → Fq → 0. Hence we have Hq(Ybq+1; Z) = H and Hq+1(Ybq+1; Z) = Ker(dq+1) = Fq+2. Let us now consider the exact homology sequence of the inﬁnite cyclic covering pq+1 : Ybq+1 → Yq+1. We have

t−1 pq+1 · · · → Fq+2 = Hq+1(Ybq+1; Z) −→ Fq+2 = Hq+1(Ybq+1; Z) −→ Hq+1(Yq+1; Z) t−1 → H = Hq(Ybq+1; Z) −→ H = Hq(Ybq+1; Z) → · · · .

By hypothesis, H is a ZT-module of type K. Hence (t − 1) : Hq(Ybq+1, Z) → Hq(Ybq+1, Z) is an isomorphism. We thus get an exact sequence:

t−1 pq+1 Fq+2 = Hq+1(Ybq+1, Z) −→ Fq+2 = Hq+1(Ybq+1, Z) −→ Hq+1(Yq+1; Z) → 0. (?)

Since Coker((t − 1) : ZT → ZT) is isomorphic to the integers Z the exact sequence is isomorphic to (ZT)rq+2 → (ZT)rq+2 → Zrq+2 → 0.

Claim. The Hurewicz homomorphism πq+1(Yq+1) → Hq+1(Yq+1; Z) is onto.

Proof of the claim. The Hurewicz homomorphism is equal to the composition of the following homomorphisms:

πq+1(Yq+1) → πq+1(Ybq+1) → Hq+1(Ybq+1; Z) → Hq+1(Yq+1; Z). 50 6 Knot modules

The ﬁrst homomorphism is the inverse of the isomorphism on homotopy groups induced by the covering projection. The second homomorphism is onto by the second part of the Hurewicz theorem. The third homomorphism is onto by the sequence (?) above (it is here that the hypothesis “H is of type K” is used).

{ 1 2 rq+2 } Let q+2, q+2,... , q+2 be a free basis of Fq+2, and consider the element ( ( j )) ∈ ( Z) ( ) pq+1 dq+2 q+2 Hq+1 Yq+1; . Since the Hurewicz homomorphism πq+1 Yq+1 → Hq+1(Yq+1; Z) is onto, we can attach to Yq+1 a (q + 2)-cell along the element ( ( j )) ( ) pq+1 dq+2 q+2 . We do this for j = 1, 2,..., rq+2 and thus obtain a q + 2 - ( ( j )) dimensional complex Yq+2. Note that the elements pq+1 dq+2 q+2 for j = 1, 2,... , rq+2 constitute a basis of the free abelian group Hq+1(Yq+1; Z). The complex Yq+2 is the complex Y we are looking for. Indeed, by construction the ZT-chain complex of Ybq+2 is isomorphic to 0 → Fq+2 → Fq+1 → Fq → 0 (far from dimensions 0 and 1). Therefore the homology modules of Ybq+2 are as announced.

Remark. Our proof follows the same lines as Kervaire’s, with some improvements brought by Levine.

(1) The spineY replaces Kervaire’s direct construction by surgery of the manifold M, which is then surgerized on the generator of the fundamental group to produce the n-knot.

(2) Kervaire constructs an n-knot with ﬁrst nontrivial knot module isomorphic to H. Since he does not use homological dimension nor the fact that projective ZT-modules are free, he cannot claim that this knot has only one nontrivial knot module (see the Paris paper [64, p. 243, lines 4 to 9]). But as shown by Levine in “Knot modules” (see [82, Lemma 9.4]), it is true that this knot has only one nontrivial knot module.

6.4 Seifert hypersurfaces

Let Kn ⊂ Sn+2 be an n-knot. We denote by N(K) a closed tubular neighborhood of Kn and by E(K) its exterior. Moreover, we choose an oriented meridian m on the boundary of E(K).

Deﬁnition 6.16. A Seifert hypersurface of Kn is an oriented and connected (n + 1)- submanifold Fn+1 of Sn+2 which has Kn as oriented boundary.

See Figures F.6 and F.7. 6.4 Seifert hypersurfaces 51

Basic facts. (I) An n-knot always has Seifert hypersurfaces. This basic property of n-knots was well-known to the topologists in the 1960s. In AppendixB, we give a proof of the existence of Seifert hypersurfaces in a more general situation. To obtain a Seifert hypersurface Fn+1 of Kn, we prove the existence of a diﬀerentiable map ψ : E(K) → S1 such that ψ is of degree +1 on m and such that the restriction ψ0 of ψ on bN(K) extends to a ﬁbration ψ˜ : N(K) → B2 where Kn = ψ˜ −1(0). We choose a regular value z of ψ. We use the notation F = (ψ−1(z)) and K 0 = bF. By construction K 0 is isotopic to Kn inside N(K). We can choose in N(K) a collar C such that bC = Kn ∪ (−K 0). By construction Fn+1 = F ∪ C is a Seifert hypersurface of Kn. But, to study E(K) it is more convenient to consider its deformation retract F = Fn+1 ∩ E(K). We also say that F is a Seifert hypersurface of Kn. (II) As F is oriented, F has a trivial normal bundle. Hence, we can choose an embedding α : I × F → E(K) where I = [−1, +1], such that the image α(I × F) = N(F) is an oriented compact tubular neighborhood of F in Sn+2 and F = α({0} × F). We use the following notation: F+ = α({+1} × F) (resp. F− = α({−1} × F)). If X ⊂ F let i+(X) = α({+1} × X), (resp. i−(X) = α({−1} × X)). So we obtain i+ : F → E(F) resp. i− : F → E(F) .

Let the exterior of F be deﬁned as E(F) = E(K)\ α(( ] − 1, +1[ ) × F). As the boundary of F is a homotopy sphere, Poincaré duality for F, followed by Alexander duality for F and E(F), implies the following proposition (very often used by Kervaire).

Proposition 6.17. Let 1 ≤ j ≤ n. Then Hj(F, Z) and Hj(E(F), Z) are isomorphic.

(III) By construction, F enables us to obtain a presentation of knot modules as follows. The map ψ induces the abelianization epimorphism of the fundamental group ρ : π1(E(K)) → Z. Let p : Eb(K) → E(K) be the covering associated to ρ. By definition, p is the infinite cyclic covering associated to Kn. By pullback we obtain a differentiable map ψb : Eb(K) → R such that ψ ◦ p = exp ◦ ψ,b where exp is the exponential map. Then there exists a regular fiber Fb of ψb that is diffeomorphic by p to F. As p−1(E(F)) has infinitely many connected copies, we choose one of them and denote it by E0(F). The others are obtained from E0(F) by the action of 52 6 Knot modules

the Galois group T isomorphic to Z. The restriction p0 of p on E0(F) is a ( ) −1 ( ) → ( ) diﬀeomorphism on E F . Then p0 : E F E0 F induces an embedding bi : E(F) → Eb(K).

We will also denote by i+, i−, bi the homomorphisms induced on the homology groups by the maps i+, i−,bi. Levine presents knot modules with the help of the following theorem (see [82, 14.2]). Theorem 6.18. Let 0 < j < n + 1, the following sequence is an exact sequence of ZT-modules:

dj ιj 0−→Hj(F, Z) ⊗ ZT −→ Hj(E(F), Z) ⊗ ZT −→ Hj(Eb(K); Z) → 0, (∗) where, for x ∈ Hj(F, Z), y ∈ Hj(E(F), Z, and a(t) ∈ ZT, we have dj(x ⊗ a(t)) = (i+(x) ⊗ ta(t)) − (i−(x) ⊗ a(t)) and ιj(y ⊗ a(t)) = a(t)(bi(y)). We now introduce Levine’s concept of simple knots. In [81] he proves the following theorem. Theorem 6.19. Let 1 < q ≤ (n + 1)/2. If an n-knot Kn ⊂ Sn+2 is such that 1 n n+1 πi(E(K)) = πi(S ) for 1 ≤ i < q, then K has a Seifert hypersurface F that is (q−1)-connected. Moreover, we can ﬁnd a (q−1)-connected Seifert hypersurface such that the embeddingsbi ◦i+, (resp.bi ◦i−) induce injective homomorphisms Hq(F, Z) → Hq(Eb(K); Z). Such an F is said to be minimal. Deﬁnition 6.20. When n is even let q = n/2; when n is odd let q = (n + 1)/2. An n n+2 1 n-knot, K ⊂ S , such that πi(E(K)) = πi(S ) for i < q is a simple n-knot. Comments. When n = 1 or n = 2, all n-knots are simple. When 2 ≤ q and n = 2q−1, the above theorem implies that we can choose a (q−1)-connected Seifert hypersurface F2q for K2q−1. In this case, F2q has the homotopy type of a bouquet of q-spheres. If 3 ≤ q, this implies, as explained in AppendixD, that such an F2q is a q-handlebody.

6.5 Odd-dimensional knots and the Seifert form

Let K2q−1 ⊂ S2q+1 be an odd-dimensional knot. We denote by N(K) a closed tubular neighborhood of K2q−1 and by E(K) its exterior. We choose a Seifert hypersurface F2q of K2q−1 and we consider F = F2q ∩ E(K). We keep the notation of Section 6.4. Let us consider the free Z-module H = (Hq(F; Z))/Tors, where Tors is the Z- torsion subgroup of Hq(F; Z). We also denote by i+ (resp. i−) the homomorphism induced by i+ (resp. i−) on the q-cycles of F. 6.5 Odd-dimensional knots and the Seifert form 53

For (x, y) in H × H, we choose representative q-cycles x0 and y0. We define the Z-bilinear form A : H × H → Z 0 0 0 0 by A(x, y) = LS2q+1 (x , i+(y )) where LS2q+1 (x , i+(y )) is the linking number between 0 0 2q+1 the two q-cycles x and i+(y ) in S as defined in AppendixA: Definition 6.21. The bilinear form A is the Seifert form associated to F2q. See Figure F.8.

The Seifert form determines the intersection form. Let I : H × H → Z be the T intersection form on H = (Hq(F; Z))/Tors. Let A be the transpose of A deﬁned as T 0 0 A (x, y) = A(y, x) = LS2q+1 (y ; i+(x )). The following formula is often quoted up to sign in the literature: (−1)qI = A + (−1)qAT . We brieﬂy present the main steps of the proof, since we shall encounter several useful formulas along the way. We have

0 0 0 0 LS2q+1 (x , i−(y )) = LS2q+1 (i+(x ), y ) by an easy translation argument,

0 0 q+1 0 0 LS2q+1 (x , i−(y )) = (−1) LS2q+1 (y , i+(x )) by the symmetry of the linking number, and

q+1 0 0 q+1 T (−1) LS2q+1 (y , i+(x )) = (−1) A (x, y) by deﬁnition of AT . Hence,

0 0 0 0 q T LS2q+1 (x , i+(y )) − LS2q+1 (x , i−(y )) = A(x, y) + (−1) A (x, y).

0 0 0 0 Lemma 6.22. The left-hand side LS2q+1 (x , i+(y )) − LS2q+1 (x , i−(y )) of the last equality is equal to (−1)qI(x, y). Proof. We have

0 0 0 0 0 0 0 LS2q+1 (x , i+(y )) − LS2q+1 (x , i−(y )) = LS2q+1 (x , i+(y ) − i−(y )) 0 = IS2q+1 (x , C),

0 where IS2q+1 (·, ·) is the intersection number between the q-cycle x and a (q+1)-chain 0 0 C such that ∂C = i+(y ) − i−(y ). 54 6 Knot modules

We claim that 0 q IS2q+1 (x , C) = (−1) I(x, y). Here is why. Observe that the intersection points in S2q+1 between x0 and C are the intersection points in F between x0 and y0. Let P be one of them. We have to compare the contribution of P to both intersection numbers. Let (e1,..., eq) be a frame at P 0 0 representing the orientation of x and let (1, . . ., q) be one for y . The contribution of P to I(x, y) compares the frame (e1,..., eq, 1, . . ., q) to a frame ( f1,..., f2q) at P that represents the orientation of F. Now C can be chosen to be equal to I × y0 0 0 0 since ∂C = i+(y ) − i−(y ). Hence the contribution of P to IS2q+1 (x , C) compares the frame (e1,..., eq, I, 1, . . ., q) to the frame (I, f1,..., f2q). To do that we have to move I in (e1,..., eq, I, 1, . . ., q) into the ﬁrst place. This introduces the factor (−1)q.

Corollary 6.23. If A is a matrix of a Seifert form associated to a (2q − 1)-knot, the determinant of (A + (−1q)AT ) is equal to ±1.

Proof. As a knot is a homotopy sphere, the intersection form (−1)qI is unimodular (i.e., its determinant is equal to ±1). The equality (−1)qI = A + (−1)qAT implies the corollary.

Deﬁnition 6.24. A square matrix A with integral coeﬃcients is a (−1)q-matrix if A + (−1)q AT is unimodular.

Proposition 6.25. Let E be a Z-basis of H = (Hq(F; Z))/Tors. Let A be the matrix of the bilinear form A in the chosen basis E and AT be the transpose of A. Then 0 there exists a basis of H = (Hq(E(F); Z))/Tors, such that, in the chosen basis, A is (q+1) T the matrix of i+ and ((−1) A ) is the matrix of i−.

Proof. In the homology exact sequence of the pair (N(F) ⊂ S(2q+1)), the connecting homomorphism (2q+1) δ Hq+1(S , N(F); Z) −→ Hq(N(F); Z) is an isomorphism. Let δ−1 be its inverse. Let ex be the excision isomorphism and let D be Poincaré duality. We consider the sequence of isomorphisms

(2q+1) ex D q Hq+1(S , N(F); Z) −→ Hq+1(E(F), bE(F); Z) −→ H (E(F); Z).

Let H00 be the free abelian group (Hq(E(F); Z))/Tors. The isomorphism D ◦ex◦ δ−1 induces an isomorphism α : H → H00. On the other hand, the universal coeﬃcient homomorphism induces an isomorphism u where

00 0 u : H → HomZ(H ; Z). 6.5 Odd-dimensional knots and the Seifert form 55

0 0 0 Let (x, y) ∈ (H × H) and let us also write i+(y ) for the class of i+(y ) in H . By 0 0 0 construction A(x, y) = LS2q+1 (x , i+(y )) = ((u ◦ α)(x))(i+(y )), where is a sign equal to 1, (−1)(q+1), or (−1)q, according to the conventions of sign used in dualities. One can choose a basis for H00 such that the matrix of α is equal to I, where I is the identity matrix. We take the dual basis for H0, and then I is the matrix of u. Hence, 0 with the chosen basis, A is the matrix of i+ : H → H . q+1 (q+1) T As LS2q+1 (x, i−(y)) = (−1) LS2q+1 (y, i+(x)), we have that ((−1) A ) is the matrix of i− in the same chosen basis.

Theorem 6.18 and the above proposition imply the following corollary.

Corollary 6.26. The matrix (At+(−1q)AT ) is a presentation matrix of the ZT-module (Hq(Eb(K); Z))/Tors, where Tors is the Z-torsion of Hq(Eb(K); Z). Of course a knot has many Seifert hypersurfaces;, therefore Seifert forms are not unique. But, Levine [81] proved that the Seifert forms of a (2q − 1)-knot, 1 ≤ q are related by an equivalence relation called S-equivalence:

Definition 6.27. We consider matrices with integral coefficients. For any 1 × n line matrix u, any n × 1 column matrix v, and any integral number x, the following matrix A0 is an elementary enlargement of a square n×n matrix A, and A is an elementary reduction of A0: 0 0 0 0 © ª A = 1 x u® . 0 v A « ¬ Definition 6.28. Two (−1)q-matrices are S-equivalent if they can be related by a chain of unimodular isomorphisms, elementary reductions, and elementary enlargements.

Theorem 6.29 (Levine [81]). Let 1 ≤ q. Two Seifert matrices associated to the same (2q − 1)-knot are S-equivalent.

Deﬁnition 6.30. Let A be a Seifert matrix of a knot Kn ⊂ Sn+2. “The” Alexander n n+2 q T polynomial ∆A(t) ∈ ZT of K ⊂ S is the determinant of (At + (−1 )A ). Corollary 6.31. If A and A0 are two S-equivalent Seifert matrices, the polynomials ∆A(t) ∈ ZT and ∆A0 (t) ∈ ZT are equivalent modulo multiplication by a unit of ZT. q T As ∆A(1) is the determinant of (A + (−1 )A ), which is equal to ±1 (see Corol- lary 6.23), the polynomial ∆A(t) is never equal to 0. We can obtain, after a possible multiplication of ∆A(t) by a unit of ZT, a unique polynomial ∆K (t) with integral coeﬃcients such that ∆K (0) is a positive integer. We can choose ∆K (t) to be the Alexander polynomial of Kn ⊂ Sn+2. An algebraic study of S-equivalence is made by Hale Trotter in [125]. 56 6 Knot modules

6.6 Even-dimensional knots and the torsion Seifert form

Let X be a (finite) CW-complex. We denote by Hj(X) the homology group Hj(X; Z), by Tj(X) the Z- torsion subgroup of Hj(X), and by Fj(X) the Z-torsion-free quotient Hj(X)/Tj(X). In his Theorem II.2, Kervaire gave necessary and sufficient conditions on a ZT- module to be the qth knot module of a simple 2q-knot. These conditions are expressed via a presentation matrix, derived from a minimal Seifert hypersurface. In this sense they are not intrinsic. Later, in “Knot modules” ([82]), Levine coined intrinsic conditions in terms of Blanchfield duality, adequately reinterpreted. It follows from both Kervaire and Levine work that there are no more conditions on the torsion-free part Fq Eb(K) besides the usual ones on a knot module (finite type and multiplication by (1 − t) is an isomorphism). The novelty concerns Tq Eb(K). Kervaire shows that it has a presentation matrix that looks formally as the presentation matrix for Fq Eb(K) of a simple (2q − 1)-knot. Here is Kervaire’s original statement of his Theorem II.2 (with minor changes to adapt notation). Theorem 6.32. Let H be a ZT-module and q be an integer ≥3. There exist a knot 2q 2q+2 1 2q+2 2q 2q+2 2q K ⊂ S with πi(S ) πi(S \ K ) for i < q and H πq(S \ K ) if and only if multiplication by (1 − t) is an isomorphism of H and H possesses a presentation of the form Í x1,..., xα; j(taij − bij)xj, dj xj where the integers dk ∈ Z with 1 ≤ k ≤ α and the square integer matrices A = (aij) and B = (bij) satisfy the relation q+1 diaij + (−1) dj bji = 0 for all couples of indices (i, j) with 1 ≤ i ≤ α and 1 ≤ j ≤ α. (Some dk are allowed to be equal to 0. They correspond to the torsion-free part of the module.)

We now tell the story of Tq Eb(K). This is the part of the module where dk , 0 and where the relation between the matrices A and B matters. Let Mm be a closed (connected) oriented manifold of dimension m. For each integer i such that 0 ≤ i ≤ m, there is a pairing

TIi : Ti(M) × Tm−i−1(M) → Q/Z defined as follows. Let x ∈ Ti(M) and let z ∈ Tm−i−1. Represent x, resp. z by disjoint cycles ξ, resp. ζ. Suppose that x is of order d. Let γ be a chain (with integer coefficients) of dimension (i + 1) such that ∂γ = dξ. Then 1 TI (x; z) = (γ ζ) i d 6.6 Even-dimensional knots and the torsion Seifert form 57 where denotes the integral intersection number. The main properties of the maps TIi are (0) they are well defined with values in Q/Z; (1) they are Z-bilinear; (2) they are nondegenerate, meaning that the adjoint ] TIi : Ti(M) → HomZ(Tm−i−1(M); Q/Z)' Tm−i−1(M) is an isomorphism; mi+1 (3) TIi = (−1) TIm−i−1. A special case takes place when m = 2q + 1 and i = q. Consider

TIq : Tq(M) × Tq(M) → Q/Z. q+1 This is a bilinear, nondegenerate, (−1) -symmetric form on the ﬁnite group Tq(M). It is usually called the “linking form” on the torsion group Tq(M). We prefer to think of it as a torsion intersection form, keeping the name “linking” for the coupling presented below. See also Wall [132]. These facts were discovered by several mathematicians in the 1920s, including Veblen, Alexander, and de Rham. A classical reference is Seifert–Threllfall’s book [111, Section 77]. They made Poincaré’s duality on torsion subgroups more precise. Of course, all this was known to Kervaire when he addressed the question of the structure of the qth knot module of a simple 2q-knot. He used the technique, without mentioning the underlying concepts. Lemma 6.33. Let F be a (2q + 1)-dimensional manifold diﬀerentiably embedded in S2q+2, with ∂F a homology sphere. Let E(F) be the exterior of F in S2q+2. Let j be an integer with 0 ≤ j ≤ 2q. Then Hj(F; Z) is isomorphic to Hj(E(F); Z). 2q+1−j Proof. By Poincaré duality, Hj(F) is isomorphic to H (F; ∂F), which is isomorphic to H2q+1−j(F) since ∂F is a homology sphere. By Alexander duality, 2q+1−j H (F) is isomorphic to H2q+2−(2q+1−j)−1(E(F)) = Hj(E(F)). Alexander duality yields the torsion linking forms

TLj : Tj(F) × T2q−j(E(F)) → Q/Z, where j is an integer such that 1 ≤ j ≤ 2q − 1. It is deﬁned as follows. Let x ∈ Tj(F) and z ∈ T2q−j(E(F)). Let ξ, ζ, γ, d be as above. Let η be an integral (2q− j +1)-chain in S2q+2 such that ∂η = ζ. Then 1 TL (x, z) = (γ η). j d 58 6 Knot modules

The properties of the torsion linking forms are (0) they are well defined with values in Q/Z; (1) they are Z-bilinear; (2) they are nondegenerate, meaning that the adjoint provides an isomorphism from Tj(F) to T2q−j(E(F)). We should remember that cohomology was not available before 1935. Naturally there is a definition of these pairings making use of cohomology and the universal coefficient theorem. Now we suppose that K2q is a simple 2q-knot in S2q+2 and that F is a minimal Seifert hypersurface for K, as Kervaire did. Following Gutierrez and Kojima (who formalized what Kervaire wrote) we define the torsion Seifert forms

TA± : Tq(F) × Tq(F) → Q/Z as follows. Let x and y be in Tq(F). Then

TA±(x, y) = TLq(x, i± y).

We recall that a Seifert hypersurface F for a simple 2q-knot is minimal if it is (q −1)- connected and if i± : Hq(F) → Hq(Y) are injective. Levine proved (implicitly) that such hypersurfaces exist (see “Unknotting spheres” [77, middle p. 14]). For a minimal hypersurface, the homomorphisms i± : Tq(F) → Tq(Y) are isomorphisms since they are injective between ﬁnite isomorphic groups. Hence the torsion Seifert forms are nondegenerate in this case. Kervaire proved that the torsion Seifert forms for even-dimensional knots have formally much in common with Seifert forms for odd-dimensional knots: q T T Proposition 6.34. We have the equality TA+ = (−1) TA− where TA− denotes the transposition of TA−.

Indications of the proof. Kervaire chooses a “basis” {ξi } for Tq(F) (with ξi of order di) and a dual basis {xj } for Tq(E(F)). The corresponding Seifert matrices are Í denoted (aij) for TA+ and (bij) for TA−. By deﬁnition i+(ξi) = aij xj and i−(ξi) = Í bij xj. One has aij = djTLq(ξj, i+(ξi)) and bij = djTLq(ξj, i−(ξi)). q In [64, bottom p. 248], Kervaire proves that diaij = (−1) dj bji. Hence

q di djTLq(ξj, i+(ξi)) = (−1) dj diTLq(ξi, i−(ξj)).

Both sides are integers. Dividing by di dj we get the equality, with values in Q/Z. Not surprisingly we get the next corollary. 6.6 Even-dimensional knots and the torsion Seifert form 59

q+1 T Corollary 6.35. We have the equality TA+ + (−1) TA+ = −TIq. Gutierrez (in [40]) proved that several of these results are true without assuming that the even-dimensional knot is simple. Without stating these results explicitly, Kervaire uses them to get a presentation of the qth knot module Hq(Eb(K); Z). Classical arguments provide the exact sequence (for more details see Levine’s [82, p. 43])

ψ 0 → Hq(F) ⊗ ZT −→ Hq(E(F)) ⊗ ZT −→ Hq(Eb(K); Z) → 0.

The homomorphism ψ is classically expressed as ψ(x ⊗ 1) = i+(x) ⊗ t − i−(x) ⊗ 1 and hence the two Seifert matrices are the tool to get a presentation of the qth knot module. It happens that the previous short exact sequence restricts to the short exact sequence (see [82, p. 44])

ψ 0 → Tq(F) ⊗ ZT −→ Tq(E(F)) ⊗ ZT −→ Tq(Eb(K)) → 0.

Recall that Kervaire proved that Tq(Eb(K)) is a ﬁnite group. We have proved that Tq(F) = Tq(E(F)) and an easy Mayer–Vietoris argument proves that Tq(F) = Tq(Eb(K)). We also have a short exact sequence for the torsion-free quotients:

ψ 0 → Fq(F) ⊗ ZT −→ Fq(E(F)) ⊗ ZT −→ Fq(Eb(K)) → 0.

The last short exact sequence provides a presentation for the Z-torsion-free quotient Fq(Eb(K)). The penultimate short exact sequence provides a presentation for q the Z-torsion subgroup Tq(Eb(K)) with the relation diaij = (−1) dj bji proved by Kervaire. Of course we must take into account that the “basis” elements are of ﬁnite order and add the corresponding relations. Thus the presentation of Tq(Eb(K)) is not square. But the presentation for Fq(Eb(K)) is. After having established that the qth knot module has presentations that have the form we have just described, Kervaire proved the opposite realization result. The proof extends over more than three pages ([64, pp. 249–252]). It relies much on Kervaire’s explicit knowledge of (q − 1)-connected, parallelizable manifolds of dimension (2q + 1) with boundary a homotopy sphere.

A brief look at what happened after Kervaire’s work on simple 2q-knots.

(1) Maurizio Gutierrez (in [40]) introduced Q/Z into the picture and generalized some of Kervaire’s results for dimensions diﬀerent from q. 60 6 Knot modules

(2) Jerome Levine (in [82]) in his exhaustive study on knot modules deﬁned a bilinear, ± symmetric, nondegenerate form on Tq(Eb(K)) with values in Q/Z, usually denoted by [ , ]. Among many other things he proved a realization result for torsion knot modules equipped with such a form. (3) Sadayoshi Kojima (in [71]) wrote a useful paper that clariﬁes many concepts. 7

Odd-dimensional simple links

7.1 q-handlebodies

Let K2q−1 ⊂ S2q+1 be a simple odd-dimensional knot. As explained at the end of Section 6.4, we can choose a Seifert hypersurface F2q of K2q−1 that is a q-handlebody. In this chapter we consider a more general situation. Deﬁnition 7.1. A (2q − 1)-simple link in S2q+1 is the boundary of a q-handlebody F2q embedded in S2q: bF2q ⊂ S2q+1.

We ﬁrst deﬁne a handle presentation of a q-handlebody.

q−1 q 2q−1 Deﬁnition 7.2. Let φj : (S × B )j → S , 1 ≤ j ≤ k be k disjoint embeddings. Let W2q be the quotient of 2q q q (B qj (B × B )j) q−1 q q q by the identiﬁcation x = φj(x), for all x ∈ (S × B )j = (bB × B )j. Let π be the corresponding quotient map:

2q q q 2q π : (B qj (B × B )j) → W .

q The image of π((B × {0})j) by this identification is the core Cj of the handle q q 2q q 2q Bj = π((B × B )j), and π(B qj (B × {0})j) is the skeleton σ(W) of W . A (2q)-manifold F2q diffeomorphic to such a W2q is a q-handlebody that has W2q as a handle presentation. The (q − 1)-dimensional link L in S2q−1 defined by

q−1 2q−1 L = qj(Lj = φj((S × {0})j)) ⊂ S is the attaching link of this handle presentation of F2q. Remark. A q-handlebody has the homotopy type of a “bouquet” of q-spheres and its boundary is always (q − 2)-connected. When q ≥ 2, a handle presentation is orientable because the intersection of each handle with B2q is connected (it is diﬀeomorphic to (Sq−1 × Bq)). If an oriented handle presentation W2q is embedded in S2q+1 it is parallelizable. We can identify a tubular neighborhood N(W) with the 2q 2q+1 normal interval bundle of W in S . Let i+ be a section of this bundle in the positive direction. “The” Seifert matrix of the handle presentation W2q is the matrix 62 7 Odd-dimensional simple links

2q of the Seifert form in a particular basis {cj, 1 ≤ j ≤ k} of Hq(W , Z) constructed as follows: Each connected component Lj of the attaching link of the handle presentation 2q bounds an oriented q-ball Dj in B . Let cj be the q-cycle obtained as the union of Dj with the core Cj. The corresponding Seifert matrix A = (aij) is given by aij = LS2q+1 (ci; i+(cj)). Remark. Let A be “the” Seifert matrix of a handle presentation of F2q. Assume that I = (−1)q(A + (−1)q AT ) is unimodular; then by Poincaré duality, • if q = 1, the boundary of F2 is S1 and bF2 ⊂ S3 is a classical knot; • if q = 2 the boundary of F2q is a homology sphere; • if q ≥ 3 the boundary of F2q is a homotopy sphere and bF2q ⊂ S2q+1 is a knot.

7.2 The realization theorem for Seifert matrices

Theorem 7.3 (Realization). Suppose that q ≥ 2. Let A be a square (k × k) matrix with integral coefficients. Then there exists some oriented handle presentation W2q embedded in S2q+1 that has A as Seifert matrix. Comments. Kervaire in [64] states this result as Théorème II.3, p. 235 and proves it in pp. 255–257. In his proof, Kervaire constructs first an abstract handlebody, essentially by following the data given by the intersection form. He then embeds the handlebody without control in S2q+1. Finally he modifies the embedding in order to realize the Seifert matrix. Levine refers to Kervaire. Here, we propose a different approach. We construct the handlebody directly in S2q+1, following step by step the data given by the Seifert matrix. This construction works for any integral square matrix A. And, when q ≥ 3, the boundary of the obtained embedded q-handlebody is a knot if and only if I = (−1)q(A + (−1)q AT ) is unimodular. Moreover, if I = (−1)q(A + (−1)q AT ) is unimodular, the construction given in the following proof also works when q = 1.

Proof of the realization theorem. We have an (r × r)-matrix A = (aij), aij ∈ Z and the matrix I = (−1)q(A + (−1)q AT ). 0 q−1 2q−1 First step. We choose r disjoint embeddings φj : (S ) → S such that

q q (−1) (aij + (−1) aij) = LS2q−1 (Li; Lj),

0 q−1 0 q−1 where 1 ≤ i < j ≤ r, Li = φi(S ), Lj = φj((S ). 7.2 The realization theorem for Seifert matrices 63

0 0 q−1 2q−1 We choose embeddings φj such that Lj = φj(S ) ⊂ S is a trivial knot 2q (this is automatic if q , 2). Each Lj bounds a diﬀerential ball Dj in B . We have

q q (−1) (aij + (−1) aij) = LS2q−1 (Li; Lj) = IB2q (Di; Dj), 1 ≤ i ≤ j ≤ r.

Second step. Let S2q be the boundary of some (2q + 1)-ball embedded in S2q+1. 0 Let i+ be a normal vector field defined on S and pointing to the exterior of the ball. We identify the sphere S2q−1 that contains the attaching link 0 q−1 L = qj(Lj = φj(S )) with an equator of the sphere S. This equator divides S into two hemispheres, the north hemisphere denoted by B and the south hemisphere denoted by B0. 2q 0 We choose identifications of B with B and B . We thus obtain ((∪j Dj) ⊂ B) 0 0 0 0 and ((∪j Dj) ⊂ B ) where Dj are the images, in B , of Dj by the chosen 0 0 identifications. Using the normal vector field i+ we push the Dj slightly 0 2q+1 0 outside B in S (keeping them fixed on Lj) getting disjoint balls Cj. Each 0 0 2q+1 0 union cj = Dj ∪ Cj is a q-sphere embedded in S . Let N(cj) be a tubular 0 2q+1 0 neighborhood of cj in S such that the N(Cj) are disjoint. The normal vector 0 0 0 field i+ restricted on the cores Cj, is a nowhere-vanishing section of N(Cj). 0 0 0 0 Let Bj be its orthogonal complement in N(Cj). The union W = B ∪j Bj is an oriented q-handlebody embedded in S2q+1 that has a presentation with cores 0 0 0 q Cj. Moreover, b(N(cj)) is diffeomorphic to cj × S . The handlebody W 0 is essentially flat, hence it is not the one we are looking for 0 0 0 to realize the matrix A. In particular, we have LS2q+1 (Dj ∪Dj; i+(Dj ∪Dj)) = 0 0 0 0 0 0 by construction. This implies that A(cj, cj) = LS2q−1 (cj; i+(cj)) = 0. Third step: Realization of the nondiagonal coefficients. From the embedded 0 2q+1 0 W ⊂ S we keep only the skeleton σ1 = B ∪ (qjCj). For 1 ≤ i < j ≤ r, let 0 0 0 0 0 aij = LS2q−1 (ci ; i+(cj)) and let bij = a1j − aij. We denote by C1j a well-chosen 0 connected sum of Cj with the boundary of b1j distinct fibers of the normal ( 0) 0 × q+1 bundle N C1 (which is isomorphic to C1 B ). These fibers are oriented by the sign of b1j. We perform the connected sum cautiously in such a way ∪ 0 ∪ that the new skeleton σ2 = B C1 2≤j ≤r C1j has disjoint cores and has new cycles c1j = Dj ∪ C1j that satisfy

0 0 LS2q−1 (c1; i+(c1j)) = a1j, 2 ≤ j ≤ r. 0 0 We also denote by i+ any extension of the positively oriented section i+, of the normal bundle of B to the new cores C1j. The indeterminacy in the choice of these extensions does not aﬀect the value of the nondiagonal coeﬃcients of A. 64 7 Odd-dimensional simple links

By the same construction, we obtain for all i, 2 ≤ i ≤ (r − 1) such a skeleton σi+1. The cores of σi+1 are constructed inductively from the cores C(i−1)j of ··· 0 σi such that Ci1 = C(i−1)1 = = C11 = C1, Ci2 = C12, ... , Cii = C(i−1)i and, for i < j ≤ r, Cij are a connected sum of C(i−1)j with the boundary of 0 bij = aij − aij ﬁbers of the normal bundle N(Cii). At the end of this step, we have a skeleton σr with a basis of q-cycles cr j = Dj ∪ Cr j, 1 ≤ j ≤ r.

We ﬁx the skeleton σ = σr , and we simplify the notation as follows: Cj = Cr j and cj = Dj ∪ Cj. The construction of σ implies

0 LS2q−1 (ci; i+(cj)) = aij, 1 ≤ i < j ≤ r. (∗)

Fourth step: Realization of the diagonal coeﬃcients. The normal bundle of cj in 2q+1 2q+1 S is trivial. We identify a tubular neighborhood N(cj) of cj in S with 0 the associated normal disc bundle. The restriction of i+ on cj is a nowhere- 0 vanishing section of this bundle. Let aj j be deﬁned as

0 0 aj j = LS2q−1 (cj; i+(cj)), 1 ≤ j ≤ r.

0 We modify i+ (only on Cj) to obtain a new section i+ such that

aj j = LS2q−1 (cj; i+(cj)), 1 ≤ j ≤ r.

Let Bj be the orthogonal complement to i+ in N(Cj). The handle presentation

2q W = B ∪j Bj

has A as Seifert matrix.

2q Remarks. In the handlebody W = B ∪j Bj, each connected component Lj of the attaching link (L ⊂ bB) has a tubular neighborhood N(Lj) = Bj ∪ bB. One identifies N(Dj) with the normal disc bundle of Dj in B and Bj with the normal disc bundle 2q of Cj in W . Both of them have a unique trivialization. The trivialization of Bj q−1 q restricted on N(Lj) gives disjoint embeddings φj : (S × B )j → bB, 1 ≤ j ≤ r, 2q which are used to define the handlebody presentation of W and its q-handles Bj . 2q Moreover, Bj ∪(B∩ N(Dj)) is isomorphic to the normal disc bundle of cj in W , which is stably trivial, but can be nontrivial. The trivializations of its restriction on Dj and Cj can be glued on Lj. This gluing is well defined, up to isomorphism, by an element of πq−1(SOq). When q is even, this element is given by the Euler number, which is equal to I(cj; cj) = 2 × A(cj, cj) = 2 × LS2q−1 (cj; i+(cj)). 7.3 Levine’s classification of embeddings of handlebodies in codimension 1 65

7.3 Levine’s classiﬁcation of embeddings of handlebodies in codimension 1

Theorem 7.4 (Levine). Let F2q be a q-handlebody embedded in S2q+1 and let A be its Seifert form. If q ≥ 3, A classifies the embedding F2q ⊂ S2q+1 up to orientation- preserving diffeomorphism of the pair F2q ⊂ S2q+1. Sketch of proof. Let F2q and F02q be two handlebodies embedded in S2q+1 with isomorphic Seifert forms. (1) The first step of the proof consists in showing that they have handle presentations W2q and W 02q with the same Seifert matrix A. This is implied by Wall [127] (see also Corollary D.4): Proposition 7.5. Let q ≥ 3 and let F2q be a q-handlebody. Then there is a natural 2q 2q 2q bijection between handle presentations of F and the basis of Hq(F ; Z) = πq(F ).

(2) Let L (resp. L0) be the attaching (q − 1)-link of the handle presentation of W2q 02q 0 (resp. W ). Let {cj, 1 ≤ j ≤ r} and {cj, 1 ≤ j ≤ r} be the q-cycles corresponding to the handle presentations (deﬁned as in the proof of the realization theorem). 0 0 0 0 By hypothesis, LS2q−1 (Li; Lj) = I(ci, cj) = I(ci, cj) = LS2q−1 (Li ; Lj). The following theorem of Haeﬂiger [42] in the case of (q − 1)-dimensional links implies that L and L0 are isotopic. Here we need (q − 1) ≥ 2, i.e., q ≥ 3.

Definition 7.6. A q-link with r components in S2q+1 is a differential submanifold 2q+1 L ⊂ S , where L = q1≤j ≤r Lj, the components Lj being orientation-preserving diffeomorphic to Sq. Note that the components are labeled. The next theorem is due to André Haefliger. See Section D.3. 2q+1 0 Theorem 7.7 (Haefliger [42]). Let q ≥ 2. Let L = qj Lj ⊂ S and L = 0 2q+1 qj Lj ⊂ S be two q-links with r components. There exists a diffeomorphism Φ : S2q+1 → S2q+1 such that (i) Φ is isotopic to the identity; 0 (ii) Φ(Lj) = Lj for 1 ≤ j ≤ r preserving both orientations; if and only if condition (♥) is satisfied:

0 0 LS2q−1 (Li; Lj) = LS2q−1 (Li ; Lj) for 1 ≤ i < j ≤ r. (♥)

0 2q+1 (3) The q-cycles cj and cj are q-spheres embedded in S . By hypothesis,

q 0 0 (−1) aij = LS2q−1 (ci; i+(cj)) = LS2q−1 (ci ; i+(cj)), i , j. 66 7 Odd-dimensional simple links

We can adapt the theorem of Haeﬂiger to obtain that the skeletons σ(W) and σ(W 0) are isotopic in S2q+1. Here, we need q ≥ 2.

(4) Point (3) implies that we can assume that σ(W) = σ(W 0) ⊂ S2q+1. Then the 0 q 2q 02q q-cycles cj = cj are embeddings of S in W and W . It is suﬃcient to 2q 02q prove that the normal disc bundles of cj in W and in W are isotopic in S2q+1. As the handle presentations are embedded in S2q+1, the bundles are stably 2q+1 trivial. Let N(cj) be a tubular neighborhood of cj in S . It is isomorphic 2q+1 to the trivial normal disc bundle of cj in S . The boundary of N(cj) is q q 0 diﬀeomorphic, via any trivialization of N(cj), to (S × S ). But cj = cj and 0 0 0 0 LS2q+1 (cj; i+(cj)) = LS2q+1 (cj; i+(cj)). This implies that i+(cj) and i+(cj) are 2q 02q isotopic on bN(cj). As the normal disc bundle of cj in W (resp. in W ) is the 0 orthogonal supplement to i+(cj) (resp. to i+(cj)) in N(cj), we have obtained the ambient isotopy between W2q and W 02q.

The abstract structure of parallelizable handlebodies is described in AppendixD. The Seifert form provides an easy way to get it when the handlebody is embedded. In particular, when 3 ≤ q the Seifert form determines the diﬀeomorphism class of the boundary of a q-handlebody. When bW2q is a homotopy sphere, we give a study of its diﬀerential type in AppendixD. Let QA : H(W2q; Z) → Z be the quadratic form associated to the Seifert form, A( ) ( ) A i.e., Q x = A x, x , and let Q2 be its reduction modulo 2.

Theorem 7.8. Let F2q be a q-handlebody embedded in S2q+1 and let A be its Seifert (− )q( (− )q T ) A form, I = 1 A+ 1 A its intersection form, and Q2 the associated quadratic form.

(1) If q is even and q > 3, then I classiﬁes F2q up to diﬀeomorphism.

(2) If q = 1, 3, 7, then I classiﬁes F2q up to diﬀeomorphism.

A 2q (3) If q is odd and q , 1, 3, 7, then I and Q2 classify F up to diﬀeomorphism.

The proof of the theorem results from what is presented in AppendixD (on handlebodies) up to one argument. We need to use the lemma on normal bundles of A A Kervaire–Vasquez [68, p. 512] in order to know that Q and Q2 coincide with the forms QW defined in AppendixE; see Definition E.4. 7.4 Levine’s classification of simple odd-dimensional knots 67

7.4 Levine’s classiﬁcation of simple odd-dimensional knots

Levine’s classification of embeddings of handlebodies in codimension 1 implies that their boundaries are isotopic. When 3 ≤ q it implies a classification theorem for simple fibered links (see Theorem 9.9) because the fibration of the exterior of the link produces a unique (up to isotopy) Seifert hypersurface with a unimodular Seifert form. In [81], Levine uses that the boundary of the Seifert hypersurfaces are homotopy spheres in order to obtain the S-equivalence relation (see Section 6.5) between the Seifert forms associated to the same knot. Today there is no available generalization of S-equivalence to links. In [81], Levine proves the following theorem: Theorem 7.9. If q ≥ 2, two simple knots are isotopic if and only if they have S-equivalent Seifert forms. When 3 ≤ q, Levine uses his classification of embeddings of handlebodies in codimension 1 and the S-equivalence relation (so, the boundary of the handlebodies has to be homotopy spheres). When q = 2, Levine needs to know that K3 is diffeomorphic to S3 to apply an important result by Wall on closed simply connected 4-manifolds. Note. The classification of simple 2q-knots is difficult, much more so than the classification of simple (2q − 1)-knots. It was performed by Cherry Kearton (see [58]), except for some difficulties with the 2-torsion. It was thoroughly treated by Michael Farber; see [34].

Knot cobordism

8.1 Deﬁnitions

Let Kn ⊂ Sn+2 be an n-dimensional knot. We denote by −Kn ⊂ −Sn+2 the knot obtained after taking the opposite orientation on Kn and on Sn+2. Let I be the interval [0, +1]. n ⊂ n+2 n ⊂ n+2 Definition 8.1. Two knots K0 S and K1 S are cobordant if there exists a smooth (n + 1)-dimensional oriented manifold Cn+1 embedded in I × Sn+2 such that n+1 − n ∪ n n+1 (1) the boundary of C is equal to K0 K1 and C meets the boundary of n+2 n+2 n+2 I × S orthogonally, with −K0 ⊂ {0} × S and K1 ⊂ {1} × S ; (2) Cn+1 is diffeomorphic to I × Σn, where Σn is the homotopy sphere diffeomorphic to K0 and K1. By definition, a null-cobordant knot is a knot cobordant to the trivial knot. Remark. To be cobordant is an equivalence relation on the set of n-knots. Moreover, the connected sum operation induces an abelian group structure on the set of cobordism classes of n-knots. The trivial knot represents the zero element and the class of −Kn ⊂ −Sn+2 is the inverse of the class of Kn ⊂ Sn+2. Definition 8.2. The standard sphere Sn+2 is the boundary of the standard ball Bn+3. A knot Kn ⊂ Sn+2 is a slice if Kn bounds, in Bn+3, an oriented manifold ∆n+1, diffeomorphic to the standard ball Bn+1, and ∆n+1 meets Sn+2 orthogonally. It is obvious that a knot is a slice if and only if it is null-cobordant.

8.2 The even-dimensional case

In his paper [64], Kervaire considers the group C2q of the cobordism classes of knots K2q ⊂ S2q+2, when K2q is diffeomorphic to the standard sphere S2q. Remark. If 2q , 4, an embedded homotopy sphere K2q ⊂ S2q+2 is diffeomorphic to the standard 2q-sphere. When 2q = 4, Kervaire needs that K4 is diffeomorphic to S4. We will see why in the sketch of proof below. 70 8 Knot cobordism

The main contribution of Kervaire’s article, in knot-cobordism theory, is the following theorem ([64, Theorem III, p. 262]).

Theorem 8.3. For all q ≥ 1, the group C2q is trivial.

It seems that Kervaire judged the proof of C2q = {0} to be easy. Indeed, it is the case if one is willing to admit the twenty pages or so of Chapters 5, 6, and 7 of Kervaire–Milnor. These pages are needed to prove the following main statement:

Main statement. The knot K2q bounds in the ball B2q+3 a contractible manifold V2q+1 that meets S2q+2 orthogonally. The main statement implies the theorem. There are two cases: (1) If 2q ≥ 5, V2q+1 is a ball (Smale). (2) When 2q = 4, Kervaire uses that K4 is diffeomorphic to S4. In this case the manifold V5 is a priori “only” contractible. But, since K4 = bV5 is diffeomorphic to the standard sphere, V5 is in fact a ball. Indeed, we can attach a 5-ball on bV5. We get a homotopy 5-sphere Σ5. Fortunately, Kervaire–Milnor prove that Θ5 = 0. Hence Σ5 is diffeomorphic to S5. By the Cerf isotopy theorem the 5-ball attached to bV5 is isotopic to a hemisphere. Then, V5 is isotopic to the other 5 hemisphere and is diffeomorphic to B . This implies that C4 = 0. Let us remark that, if we admit homotopy spheres as 4-knots, in other words possibly exotic differential structures on S4, the cobordism group of such 4-knots could be nontrivial (if such exotic structures exist).

(3) When 2q = 2, more surgery is needed on the homotopy ball V3 (Kervaire did it in [64, Lemme III.7, p. 265]), to obtain a standard 3-ball. Kervaire’s proof does not require a proof of the Poincaré conjecture.

The proof of the main statement has two steps. (I) Let F2q+1 be a Seifert hypersurface for K2q. It is a framed diﬀerentiable manifold that has the standard sphere as boundary. Now, we can use the following theorem: Theorem 8.4 (Kervaire–Milnor [67]). There exists a framed manifold W2q+2 obtained from F2q+1 × I by attaching handles of index j ≤ q + 1 on F × {1} and such that bW = (F × {0}) ∪ (bF × I) ∪ V with V2q+1 contractible and (F × {0}) ∩ (bF × I) = bF × {0} and also (bF × I) ∩V = bF × {1} = bV. 8.3 The odd-dimensional case 71

(II) The second step of the proof consists in embedding W2q+2 in the ball B2q+3 in such a way that the image of F2q+1 × {0} is equal to the already chosen Seifert hypersurface of K2q. The image of (bF × I) ∪ V2q+1 provides the contractible manifold that has K2q as boundary.

Kervaire provided two proofs of the existence of the embedding of W2q+2. One is given in [64], the other is in his Amsterdam paper [66].

The Amsterdam paper proof. We begin by invoking Hirsch’s theory of immersions (in fact a relative version of it). Since W is framed with nonempty boundary it is parallelizable. Hence there exists an immersion of W in the ball which coincides with the identiﬁcation of F2q+1 × {0} with a Seifert hypersurface. Then we use a trick (much used by Morris Hirsch and others) that consists in jiggling the images of the cores of the handles to get them embedded and disjoint. The jiggling is possible since the highest dimension of the cores is (q +1) and since 2(q +1) < 2q +3 (general position). Since immersions constitute an open set among the diﬀerentiable maps, we get a new immersion, which is an embedding on a neighborhood of a spine of W (rel. F2q+1 × {0}) and which is therefore regularly homotopic to an embedding.

Kervaire’s proof from his Paris paper. People who prefer algebra to topology hate the jiggling arguments. Here is a sketch of a more reasonable proof. Kervaire– Milnor’s proof of the existence of the manifold W results from a careful succession of framed surgeries. Kervaire shows that these surgeries can be embedded in the ball B2q+3. Two ingredients are important: (1) In the Kervaire–Milnor approach, obstructions to perform the surgeries take place in homotopy groups that are “stable”. It happens that in order to perform embedded surgeries, the obstructions we meet are “already” stable and hence vanish, since they vanish in the nonembedded case.

(2) To embed the cores of the handles, we need a general position argument, which works as above since again 2(q + 1) < 2q + 3. Remark. The general position argument fails in the case of odd-dimensional knots since we have cores of handles of dimension q + 1 in the ball B2q+2.

8.3 The odd-dimensional case

2q−1 2q+1 Here we denote by C2q−1 the group of the cobordism classes of knots K ⊂ S , when K2q−1 is a homotopy sphere. 72 8 Knot cobordism

In contrast to the even-dimensional case, the odd-dimensional cobordism groups are nontrivial. Kervaire, who considers only the cobordism classes of knots dif- 2q−1 feomorphic to the standard S , proved that the cobordism groups C2q−1 are not finitely generated. His “easy proof” uses the Fox–Milnor condition on the Alexander polynomial. The next step was performed by Levine, who reduced the computation of C2q−1 to an algebraic problem. We summarize Levine’s theory. From the Seifert form, an obstruction is constructed for a knot to be null-cobordant (which is also valid when q = 1). An equivalent of the Witt relation can be defined on the (nonsymmetric) Seifert forms. If q > 1, the “Witt group” of Seifert forms produces a complete classification of knots in higher dimensions up to cobordism. Let us be more explicit. Definition 8.5. An -form is a bilinear form A : M × M → Z, where M is a free Z-module of finite rank r, such that I = A + AT is an -symmetric unimodular form (i.e., the determinant of I is equal to ±1). An -form is null-cobordant if r = 2r 0 is even and if there exists a submodule H of M, of rank r 0, that is pure (i.e., the quotient M/H is Z-torsion-free), and such that A(x, y) = 0 for all (x, y) ∈ H × H. Such an H is called a metabolizer of A. The unimodularity of I implies that a metabolizer H of A is equal to its orthogonal for I.

Deﬁnition 8.6. Two -forms A1 and A2 are cobordant if the orthogonal sum A1−A2 is null-cobordant. Remark. To be cobordant is an equivalence relation on the set of the -forms. The transitivity needs a proof (see Levine [80] or Kervaire [66]). The orthogonal sum provides the structure of an abelian group on the set of equivalence classes. We denote this abelian group by C . Theorem 8.7 (Levine). If 1 ≤ q, the Seifert forms of cobordant (2q − 1)-knots are cobordant (−1)q-forms. Remark. A knot has several Seifert forms since it has several Seifert hypersurfaces. In particular, the theorem implies that two Seifert forms of the same knot are cobordant. Corollary 8.8. Let = (−1)q. The correspondence that associates to each knot one of its Seifert forms induces a group homomorphism:

γq : C2q−1 → C .

One can find many proofs of this theorem. The first proof was given by Levine (see also Kervaire’s Amsterdam paper [66]). It is sufficient to prove the following statement: Null-cobordant knots have null-cobordant Seifert forms. We now give an idea of the proof of the statement. 8.3 The odd-dimensional case 73

Idea of proof of Theorem 8.7. Let F2q be a Seifert hypersurface of a (2q − 1)-knot K2q−1 ⊂ S2q+1, which bounds in B2q+2 an embedded ball ∆2q. Let us consider the closed oriented submanifold M2q = F2q ∪ ∆2q in B2q+2. As M2q is a codimension 2 embedded submanifold of B2q+2, by an argument similar to the existence of the Seifert hypersurfaces for links (see AppendixB), M2q bounds in B2q+2 an oriented subman- 2q+1 2q 2q+1 ifold W . Let jq : Hq(M , Z) → Hq(W , Z) be the homomorphism induced by the inclusion. Using the homology exact sequence of the pair (W2q+1, M2q) and Poincaré duality, one can show that Ker jq is a metabolizer of the Seifert form A associated to F2q. In any odd dimensions, the theorem also gives a necessary condition for a knot to be null-cobordant, which is readable on the Alexander polynomial. It is this argument that enables Kervaire to obtain the last main theorem of his paper [64]: “ Pour 1 ≤ q, le groupe C2q−1 n’est pas de type ﬁni”. Now, we explain this result with the help of the theorem above. Deﬁnition 8.9. Let A be a Seifert matrix of K2q−1 ⊂ S2q+1 and let P(t) = det(t A + (−1)q AT ). The Alexander polynomial of K2q−1 is the class of P(t), modulo multiplication by a unit of Z[t, t−1]. By the previous theorem, if K2q−1 is null-cobordant, any of its Seifert matrices A is null-cobordant. Then, a direct computation of det(t A + (−1)q AT ) implies a Fox–Milnor condition on the Alexander polynomial of null-cobordant knots, which is expressed in the following corollary. Corollary 8.10. Let 1 ≤ q. If P(t) is the Alexander polynomial of a null-cobordant (2q − 1)-knot, there exists a polynomial Q(t) in Z[t] such that P(t) = Q(t)Q(t−1) modulo multiplication by a unit of Z[t, t−1].

In his paper [80], Levine considers the homomorphism γq and shows the following result.

Theorem 8.11. (1) If q , 2, γq is surjective.

(2) In C+1, let C0 be the subgroup of the classes of +1-forms A such that 16 divides T the signature of A + A . Then, γ2 is an isomorphism on C0.

(3) If 2 < q, γq is an isomorphism. Remark. To obtain the surjectivity in statement (3), we need to consider knots that are homotopy spheres. Now, we give some indications of the proof of the above statements. Sketch of proof. When q ≥ 3, the realization theorem of Seifert forms [64] implies the surjectivity of γq. The ﬁrst step to proving the injectivity is the following theorem of Levine [80] (also proved by Kervaire in [66] when q ≥ 3): 74 8 Knot cobordism

Theorem 8.12. Let q ≥ 2. Every (2q − 1)-knot is cobordant to a simple knot. Let F2q be a Seifert hypersurface of a knot K2q−1. Kervaire and Milnor provide ([67, Theorem 6.6]) the following result: Theorem 8.13. Let q ≥ 2 and let F2q be a framed diﬀerentiable manifold whose boundary is a homotopy sphere. Then there exists a framed manifold W2q+1 obtained from F2q × I by attaching handles of index j ≤ q on F2q × {1} and such that

bW2q+1 = (F2q × {0}) ∪ (bF2q × I) ∪ X2q with X2q (q−1)-connected. Moreover, we have (F2q ×{0})∩(bF2q ×I) = bF2q ×{0} and bX2q = bF2q × {1}. The manifold W2q can be embedded in the ball B2q+2 in a way similar to the even-dimensional case (see Theorem 8.4). An “engulfing argument” produces an embedding of X2q in a (2q + 1)-sphere S2q+1 ⊂ B2q+2. For the use of engulfing see Levine’s paper ([80, proof Lemma 4]). A simplified use of engulfing is given by Kervaire in his Amsterdam paper [66, middle p. 91]. Then, K2q−1 is cobordant to the boundary of X2q, which is a simple knot. 2q−1 The next step considers a knot K in the kernel of γq. From the last result we can assume that K2q−1 bounds a (q−1)-connected Seifert surface F2q and that F2q has a null-cobordant Seifert form A. This implies that we can find a basis {cj, 1 ≤ j ≤ r} ∗ of a metabolizer of A that can be completed in a basis {cj, cj, 1 ≤ j ≤ r} of 2q q Hq(F , Z), which is a hyperbolic basis for the intersection form I when (−1) = +1 and symplectic when (−1)q = −1. We push F2q inside the ball B2q+2 with a small collar around K2q−1 = bF2q. Let F be the result of this push. If q ≥ 2 there is no obstruction to perform an embedded surgery on the q-cycles {cj, 1 ≤ j ≤ r} because ∗ A vanishes on the metabolizer. As the basis {cj, cj, 1 ≤ j ≤ r} is either hyperbolic or symplectic for I, the result of this surgery is diffeomorphic to a 2q-ball. The knot K2q−1 is null-cobordant.

The hard work is now to describe the group C . In [80] Levine gives a complete list of cobordism invariants of an -form. In [66] Kervaire obtains partial results on the group C and in [122] Neal Stoltzfus describes it in detail. Andrew Ranicki has constructed a thorough algebraization of higher-dimensional knot theory, including Seifert surfaces and Blanchﬁeld duality. He applies his concepts to various situations such as simple knots, ﬁbered knots, and knot cobordism. See [105, 106]. 9

Singularities of complex hypersurfaces

In this chapter we summarize some results on the topological type of singularities of complex hypersurfaces. We present results or problems related to Kervaire’s work on homotopy spheres and knots in higher dimensions (Seifert forms, knot cobordism, etc.) only.

9.1 The theory of Milnor

We now recall the main features of Milnor’s theory contained in his famous book [93]. Let f : (Cq+1, 0) → (C, 0) be a germ of a holomorphic function with an isolated critical point at the origin in Cq+1. 2q+2 We denote by Br the (2q + 2)-ball, with radius r > 0 centered at the origin q+1 2q+1 2q+2 of C and by Sr the boundary of Br . Let us use the following notation: 2q+2 −1 2q+1 −1 F0 = B ∩ f (0) and Lf = S ∩ f (0).

Basic results. −1 2q+1 0 0 (I) For a sufficiently small , f (0) meets S0 transversally for all , 0 < ≤ , 2q+2 and the homeomorphism class of the pair (F0 ⊂ B ) does not depend on such a sufficiently small . By definition, it is the topological type of f . 2q+1 (II) The orientation-preserving diffeomorphism class of the pair (Lf ⊂ S ) does not depend on such a sufficiently small . By definition, it is the link of f . By Milnor’s conic structure theorem ([93, p. 18]), it determines the topological type of f . (III) There exists a sufficiently small η, 0 < η << , such that f −1(t) meets 2q+1 2 S0 transversally for all t ∈ Bη. For such a sufficiently small η, N(Lf ) = −1 2 2q+1 f (Bη) ∩ S is a tubular neighborhood of Lf and the restriction f |N(L f ) of 2 f on N(Lf ) is a (trivial) fibration over Bη.

−1 2q+2 (IV) The manifold Ft = f (t) ∩ B for some t , 0 is the Milnor ﬁber Ff of f . Its boundary is diﬀeomorphic to the link Lf since the critical point is isolated. 76 9 Singularities of complex hypersurfaces

The Milnor ﬁber is oriented by its complex structure and Lf is oriented as its boundary.

−1 2 2q+2 (V) Moreover, N( f ) = f (Bη) ∩ B is a compact neighborhood of F0 = −1 2q+2 −1 1 2q+2 f (0)∩B . In the boundary of N( f ), we consider T( f ) = f (Sη)∩B . The conditions on and η imply that the restriction f |T(f ) of f on T( f ) is a 1 ﬁbration over Sη. We can say that T( f ) is the tube around F0, and that f |T(f ) is the tubular ﬁbration associated to f .

Open books were introduced (without the name) by Milnor in [93]. The reader will ﬁnd more details on open books in AppendixC. We use the following deﬁnition:

Definition 9.1. Let L be a closed oriented (2q−1)-dimensional submanifold of S2q+1. An open book decomposition of S2q+1 with binding L is given by a differentiable fibration ψ over the circle: ψ : E(L) → S1, and a (trivial) fibration φ: φ : N(L) → B2, such that the restrictions of ψ and φ on bN(L) = bE(L) coincide. For z ∈ S1, a fiber F = ψ−1(z) of ψ is a page of the open book. Let h : F → F be a diffeomorphism of F. The mapping torus T(F; h) of F by h is the quotient of F × [0, 1] by the gluing (x, 1) ∼ (h(x), 0) for x ∈ F.A diffeomorphism h of F is a monodromy of ψ if the projection on the second factor, (x, t) 7→ t, induces a fibration π : T(F; h) → S1 isomorphic to ψ.

Remark. The open book decomposition implies that the boundary bF of the page deﬁned above is isotopic to L in N(L), and that there exists a monodromy of ψ that is the identity on bF.

2q+1 Theorem 9.2 (Milnor ﬁbration theorem). Let E(Lf ) = S \ N˚(Lf ) be the exterior f 2q+1 of Lf . The restriction of k f k on E(Lf ) provides an open book decomposition of S with binding Lf and page Ff .

f Proof. In [93], Milnor shows that the restriction ψ of k f k on E(Lf ) is a differentiable 1 | fibration. We know (see point (III) above) that φ = η f N(L f ) is a (trivial) fibration 2 −1 1 2q+1 over B . Since bN(Lf ) = f (Sη) ∩ S , then ψ restricted on bE(Lf ) = bN(Lf ) is equal to φ.

Remark. In his proof, Milnor constructs a vector field that pushes out the tubular f fibration f |T(f ) into the fibration k f k , defined on E( f ). Both of these isotopic fibrations can be called a Milnor fibration of f . In particular, 9.1 The theory of Milnor 77

(1) the Milnor fiber is a parallelizable manifold because it is isotopic to a Seifert hypersurface of the link Lf ; (2) N( f ) is a (2q + 2)-dimensional ball with corners because the inverse of this 2q+2 −1 2 2q+2 isotopy pushes B into N( f ) = f (Bη) ∩ B . Definition 9.3. We say that N( f ) is the crooked ball of f . See Figure F.9. When f has a nonisolated singular locus at the origin, this Milnor isotopy of fibrations still works. But, if the origin is not an isolated singular point of f, the f fibration k f k does not provide an open book decomposition and in general, the boundary bFf of the Milnor fiber is no longer homeomorphic to Lf . In general Lf is not a manifold, while bFf is.

Theorem 9.4. If f has an isolated critical point at the origin, its Milnor ﬁber Ff is a q-handlebody. Proof. Without assuming that f has an isolated critical point, Milnor proves in [93, Chapter 5] that

(i) Ff has the homotopy type of a ﬁnite C.W. complex of dimension q;

(ii) Lf is (q − 2)-connected. If f has an isolated critical point, Milnor proves in [93, Chapter 6], using the existence of an open book decomposition, that

(iii) Ff is (q − 1)-connected and hence has the homotopy type of a bouquet of q-spheres;

(iv) since Lf = bFf , the boundary bFf is (q − 2)-connected. Therefore, if q ≥ 3, from Smale’s recognition theorem for handlebodies, Milnor deduces that the fiber Ff is a q-handlebody and he conjectures that it is still true when q = 2. Notice that Theorem 9.4 is trivially true when q = 1. In [76], Lê Dung˜ Tráng and Bernard Perron prove that Ff is a q-handlebody, for all q ≥ 1. Their proof, based on Lê’s carrousel construction, is independent of Milnor’s technique and they don’t use Smale’s results. For q = 2, it is a positive answer to Milnor’s conjecture. Definition 9.5. A (2q − 1)-simple fibered link L ⊂ S2q+1 is a closed oriented (2q − 1)-dimensional submanifold of S2q+1 that is the binding of an open book decomposition, the page being a q-handlebody. A simple fibered link such that L is a homotopy sphere is a simple fibered knot (also called a simple fibered spherical link).

Corollary 9.6. If Lf is the link associated to a germ f that has an isolated critical point, Lf is a simple ﬁbered link. 78 9 Singularities of complex hypersurfaces

9.2 Algebraic links and Seifert forms

Let L ⊂ S2q+1 be a simple fibered link. By definition, L is the binding of an open book decomposition that has a q-handlebody F as a page. Then Hq(F; Z) is a free Z-module of finite rank. We follow the notation and definitions of Section 6.5 for the page F. We have the maps

i+ : F → E(F)(resp. i− : F → E(F)).

0 0 Let (x , y ) be a pair of q-cycles that represent (x, y) ∈ Hq(F; Z) × Hq(F; Z). The Seifert form AF associated to F is deﬁned by

0 0 AF (x, y) = LS2q+1 (x , i+(y )).

T Let I : Hq(F; Z) × Hq(F; Z) → Z be the intersection form on Hq(F; Z). Let AF be the transposition of AF . Let us recall that, with our conventions of signs, we have

(−1)qI = A + (−1)qAT .

Proposition 9.7. (1) The Seifert form AF , associated to a page F of a simple fibered link is unimodular. (2) The fibration ψ : E(L) → S1, given by the open book decomposition, admits −1 ◦ h = i−1 i+ as a monodromy. (q+1) T −1 (3) Let A be the matrix of AF in a Z-basis B of Hq(F; Z). Then, ((−1) (A ) A) is a matrix of hq in the basis B, where hq denotes the homomorphism induced by h on Hq(F; Z). Proof. By definition, a bilinear form defined on a free module of finite rank is unimodular if it has an invertible matrix. As F is a fiber of a fibration ψ : E(L) → S1, the map i+ : F → E(F) induces an isomorphism, also written as i+, from Hq(F; Z) to Hq(E(F); Z). So, the determinant of a matrix of i+ is ±1. Proposition 6.25 implies that the determinant of a matrix of AF is equal, up to sign, to the determinant of a matrix of i+. Point (2) is obvious if we define i+ as half a turn in the positive direction and i− in the negative direction. By Proposition 6.25, point (2) implies point (3).

Corollary 9.8. Let µ be the rank of Hq(F; Z). The characteristic polynomial ∆(t) (q+1) T −1 q of the monodromy is the determinant of (tIµ − (−1) (A ) A). As (−1) I = A + (−1)qAT , we have ∆(1) = ±1 if and only if L is a homology sphere. Indeed, as F is (q − 1)-connected, L is a homology sphere if and only if the intersection form I is unimodular. As F is a q-handlebody, if q ≥ 3, it is suﬃcient for L to be a homology sphere to be also a homotopy sphere. 9.3 Cobordism of algebraic links 79

Let U(B) be the set of equivalence classes, modulo isomorphism, of unimodular bilinear forms defined on free Z-modules of finite rank. One can find the following theorem in [29].

Theorem 9.9. If q ≥ 3, to associate the Seifert form AF of the page F of a simple fibered link induces a bijective map σ between the isotopy classes of simple fibered links in S2q+1 and U(B). Sketch of proof. If q , 2, Browder’s lemma 2 (see AppendixC) implies that the Seifert forms associated to isotopic simple fibered links are isomorphic and hence σ is well defined. If q ≥ 3 Levine’s embedding classification theorem (see Theorem 7.4) implies the injectivity of σ. Kervaire’s realization theorem (see Theorem 7.3) and the h-cobordism theorem imply the surjectivity of σ.

Remark. If q = 1, the isomorphism class of the Seifert form AF associated to a page of the open book decomposition is still an invariant of the link, but σ is not an isomorphism. If q = 2, the isomorphism class of the Seifert form AF is an invariant only of the open book decomposition. Definition 9.10. A (2q − 1)-simple fibered link, L ⊂ S2q+1, is an algebraic link if 2q+1 2q+1 q+1 L ⊂ S is isotopic to a link Lf ⊂ S associated to a germ f : (C , 0) → (C, 0) of a holomorphic function with an isolated critical point at the origin in Cq+1. Remark. Algebraic links are a very special kind of simple fibered links. The question of the characterization of algebraic links among simple fibered links is an open problem. This problem is presented in detail in [87]. An important necessary condition to be an algebraic link is given by the monodromy theorem: “The monodromy hq is a quasi-unipotent endomorphism”. This necessary condition is far from being sufficient. Remark. For q ≥ 3, Theorem 7.4 implies that (2q − 1)-algebraic links are classified, up to isotopy, by the isomorphism class of the Seifert form associated to their Milnor fiber. But this result is not true if q ≤ 2. In [12], the authors construct families of pairs of germs f and g defined on (C2, 0), with isomorphic Seifert forms and nonisotopic 3 2 2 links in S . By Sakamoto [109] the germs f1 = f (x, y) + z and g1 = g(x, y) + z obtained by suspension also have isomorphic Seifert forms. In [6], Enrique Artal shows that the 3-manifolds Lf1 and Lg1 are not homeomorphic. Hence f1 and g1 do not have the same topological type.

9.3 Cobordism of algebraic links

In [10], an equivalence relation, called algebraic cobordism, is deﬁned between unimodular bilinear forms with integral coeﬃcients. As a generalization of Kervaire– 80 9 Singularities of complex hypersurfaces

Levine’s theory of knot cobordism, the following theorem is obtained: Theorem 9.11. For all q > 0, the Seifert forms associated to a page of two cobordant (2q − 1)-dimensional algebraic links are algebraically cobordant. If q ≥ 3, this necessary condition to be cobordant is sufficient. Of course this theorem can be applied to algebraic links. But, as algebraic links are “rare”, their classification up to cobordism is particular. Here are some typical results: (I) In [74], Lê shows that cobordant algebraic links in S3 are isotopic. (II) A null-cobordant (2q − 1)-algebraic knot (q > 0), is always trivial (i.e., a null-cobordant algebraic knot is never associated to a nonsingular germ). This result is proved in [87] with the help of the following result by A’Campo [1] (see also [75]): An algebraic link always has a monodromy diffeomorphism without fixed point. (III) In higher dimensions, the cobordism of algebraic links no longer implies isotopy. Explicit examples are constructed in [11].

9.4 Examples of Seifert matrices associated to algebraic links

There are methods of computation of Seifert forms associated to algebraic links in S3 (e.g., in [12]). We state Sakamoto’s formula which computes, by induction, Seifert forms associated to some higher-dimensional algebraic links. As the determination of signs is delicate, we give explicit matrices of the Seifert form, the monodromy, and the intersection form associated to a “cusp” singular point. We use the following notation: n+1 m+1 Let u = (u0,..., un) ∈ C , v = (v0,..., vm) ∈ C , and

n+1 m+1 n+m+2 w = (u0,..., un, v0,..., vm) ∈ C × C = C .

Let f : (Cn+1, 0) → (C, 0) and g : (Cm+1, 0) → (C, 0) be germs of holomorphic functions with an isolated singularity at the origin. Deﬁne h : (Cn+m+2, 0) → (C, 0) by h(w) = f (u) + g(v). Clearly h also has an isolated singularity at the origin. Let us denote by Af , Ag, Ah the respective Seifert forms, associated to the Milnor ﬁbers Ff , Fg, Fh. Remember that the complex structure provides orientations (we have no choice here). Sakamoto’s formula [109] is

(n+1)(m+1) Ah = (−1) Af ⊗ Ag. 9.5 Mumford and topological triviality 81

( ) 2 2 ··· 2 (i) Let fn u0, u1,..., un = u0 +u1 + +un be the germ for the ordinary quadratic singularity. Then the Seifert matrix for fn is equal to (1) when n ≡ 0 or 3 mod 4 and is equal to (−1) when n ≡ 1 or 2 mod 4. By point (3) of Proposition 9.7, its monodromy is equal to (−1)n+1.

(C ) → (C ) ( ) 3 (ii) Let g0 : , 0 , 0 be the germ deﬁned by g0 u0 = v0. Let F0 be the Milnor ﬁber of g0. There exists a basis of H0(F0; Z) such that the Seifert matrix A0 and the monodromy matrix h0 of g0 in this basis are

1 0 0 −1 A = , h = . 0 −1 1 0 1 −1

≥ ( ) 3 2 (iii) Let n 1. We consider the singularity given by gn u0, u1,..., un = u0 + u1 + ··· 2 2 + un−1 + un. Let Fn be its Milnor ﬁber. Using Sakamoto’s formula and the formulas given above, the Seifert matrix An, the matrix hn of the monodromy, and the matrix In of the intersection form in a chosen basis Hn(Fn; Z) can be easily computed. For example,

−1 0 0 1 0 1 A = , h = , I = , 1 1 −1 1 −1 1 1 −1 0 −1 0 0 −1 −2 1 A = , h = , I = . 2 1 −1 2 1 −1 2 1 −2

9.5 Mumford and topological triviality

We now make a return in history and present some important results in the topological theory of singularities. The paper [97] by Mumford marked the beginning of a new era. Let Σ be a normal complex algebraic subspace of dimension 2 in some Cn+1. Let P ∈ Σ be a singular or smooth point. Normality and dimension 2 imply that P has a neighborhood U such that U \ P is smooth and connected. We intersect U with a small sphere S2n+1, centered at P. The transversal intersection U ∩ S2n+1 is a 3-dimensional differentiable manifold L. It is connected, oriented, without boundary. Mumford proves carefully that the oriented diffeomorphism type of L is well defined. Let us call it the “boundary” of the singularity P ∈ Σ, since it is topologically the boundary of a cone with vertex P (for a proof of the last statement see Milnor’s book [93, Chapter 2]). The (surprising!) result obtained by Mumford is the following.

Theorem 9.12. Suppose that the point P is singular in Σ. Then L is not simply connected. 82 9 Singularities of complex hypersurfaces

Put differently, if L is simply connected, the point P is smooth. In other words, smoothness (which is an analytic concept) can be detected topologically. Mumford raised the following question. Question. What happens in higher dimensions, i.e., if the complex dimension of Σ is ≥3? One can answer the question in the case of a singular germs of hypersurfaces. First of all the boundary Lg of the Milnor fiber associated to g(z0, z1, z2, z3) = 3 2 2 2 5 z0 + z1 + u2 + u3 is diffeomorphic to S . Indeed, it is a homotopy 5-dimensional 2 sphere (the characteristic polynomial of its monodromy being ∆g(t) = t − t + 1). But, Kervaire and Milnor proved that there is only one differentiable structure on S5. 4 But g has an isolated critical point at the origin in C , and Lg is also the boundary of {g = 0}. Hence it is an example of a singular germ, which has the standard differential sphere as boundary. Hence, the theorem of Mumford cannot be generalized without some stronger concept of topological triviality. Let f : (Cq+1, 0) → (C, 0) be a germ of a holomorphic function with a (not q+1 necessarily isolated) critical point at the origin in C . The local multiplicity m0( f ) of f is the degree of the monomials of smallest degree of f ∈ C{z0,..., zq }. Clearly it is an analytic invariant. So, f is analytically trivial if and only if m0( f ) = 1. When 0 is an isolated critical point, Milnor proved that the topology of the 2q+1 embedded link Lf ⊂ S does detect regular points. This answered another question raised by Mumford. First of all, in [93], Milnor proved that the rank µ of H(Ff , Z) is equal to the dimension of the local algebra of the Jacobian ideal of f (this nontrivial result makes sense only if 0 is an isolated singular point). But the dimension of the Jacobian ideal is equal to 0 if and only if m0( f ) = 1. Therefore 0 is a regular point of f if and only if µ = 0. We can state Milnor’s [93, Corollary 7.3, p. 60] as follows: Theorem 9.13. If 0 is an isolated critical point or a regular point of f and if 2q+1 Lf ⊂ S is a trivial knot then 0 is a regular point of f . The idea is now to take the following definition of topological triviality of f , which is also valid if 0 is not an isolated singular point: f is topologically trivial if its Milnor fiber has the homotopy type of a ball. In fact it would be enough to require that the Milnor fiber has the homology of a point. With this definition, one can give the following proof that topological triviality is equivalent to analytic triviality, which is valid even if 0 is not an isolated singular point. 9.6 Joins and Pham–Brieskorn singularities 83

When m0( f ) ≥ 2, Lê constructs, by induction on q, a monodromy that has no fixed point (see [75]). So, if m0( f ) ≥ 2 the Milnor fiber does not have the homotopy type of a 2q-ball. One can also use Norbert A’Campo’s results in [2] to obtain the same result. Mumford also introduces the concept of “plumbing” to describe the topology of a good resolution of a normal singular point of a complex surface. The boundary of such 4-dimensional plumbings has been classified by Neumann in [99]. Except for very special cases (singularities that Arnold calls simple), the plumbing of a minimal good resolution and the Milnor fiber of a germ f : (C3, 0) → (C, 0) with an isolated critical point at 0 are not diffeomorphic. In particular, the Milnor fibers are always parallelizable manifolds (it is an important consequence of the Milnor fibration theorem). But in general this is not the case for the plumbings of the minimal good resolution of f : (C3, 0) → (C, 0). Moreover, Milnor fibers of germs of surfaces in C3 are always simply connected. In general this is not the case for the plumbings of the local good minimal resolution. In fact the manifold obtained by plumbing is simply connected if and only if the plumbing graph is a tree with each vertex a rational curve (i.e., a 2-sphere).

9.6 Joins and Pham–Brieskorn singularities

Deﬁnition 9.14. Let P : (Cn+1, 0) → (C, 0) be a germ of holomorphic function deﬁned by n Õ ai P(z0, z1,..., zn) = zi . i=0

If all ai are ≥2, the hypersurface Σ(a0, a1,..., an) deﬁned by P = 0 has an isolated critical point at the origin in Cn+1. Such a singularity is now called a Pham–Brieskorn singularity.

First we give early results concerning particular cases.

The case ai = 2 for all i = 0, 1,..., n has a long history. The corresponding singularity is called the ordinary double point. It was ﬁrst encountered by Emile Picard (at the end of the 19th century) and later by Salomon Lefschetz (1924). If n = 2 it was known to Poul Heegaard, in his thesis published in 1898, that the boundary of the singularity is the lens space L(2, 1), which is also known as the real projective space P3(R) and as the circle bundle associated to the tangent bundle to the 2-sphere S2 (neglecting orientations). It seems to have been known for a long time that for any n ≥ 2 the boundary of the ordinary double point is diﬀeomorphic to the sphere bundle associated to the tangent bundle of the n-sphere Sn, also known 84 9 Singularities of complex hypersurfaces

as the Stiefel manifold Vn+1,2. A proof is given by Hirzebruch [53, p. 30]. It cleverly uses the fact that the orthogonal group O(n + 1) acts on the hypersurface. The next question clearly was, what happens if we allow some ai to be ≥3? The story is told in detail by Brieskorn in his great paper [16]. See also Durfee’s paper [30], which gives precise dates for these events. While at MIT, in the winter 3 Ín 2 1965–66 Brieskorn proved that the boundary of the singularity z0 + i=1 zi = 0 is topologically a sphere of dimension (2n−1) if n is odd. So the relevant dimensions of the spheres are 5, 9, 13, ... . The proof is a clever mixture of projective geometry and algebraic topology (it uses Smale’s result that homotopy spheres are homeomorphic to the standard sphere in high dimensions).

Questions. What is the differentiable structure on this boundary sphere? For 2n−1 = 5 there is no problem since Kervaire and Milnor proved that there is only one differentiable structure on S5. But what about the sphere S9? Kervaire had proved that, in this dimension, there are exactly 2 possible differentiable structures. So, which one is it?

Brieskorn says that he talked about this question with Klaus Jänich in Cornell at the beginning of February 1966. Then Jänich wrote to Hirzebruch. Using the Ín 2 action of O(n) on the sum i=1 zi in the defining equation, Hirzebruch proved (using Jänich’s work) that for all odd values of n the boundary of the singularity defined by 3 Ín 2 z0 + i=1 zi = 0 is diffeomorphic to the boundary of the plumbing usually denoted by A2 with weights equal to 1 mod 2 (it consists of two copies of the disc bundle associated to the tangent bundle to Sn plumbed together). This plumbing is often known as the Kervaire plumbing. Kervaire proved that the boundary of this plumbing is an exotic homotopy sphere of dimension 2n −1 if and only if the Kervaire invariant vanishes in dimension 2n (remember that n is odd). When 2n = 10, Kervaire proved in his famous paper [62] that this invariant vanishes. So, Hirzebruch identified the 3 Í5 2 boundary of z0 + i=1 zi = 0 as the first exotic homotopy sphere occurring as the boundary of an isolated singularity. At the same time (spring 1966) Hirzebruch proved more (for details see Hirze- d 2 ··· 2 bruch [53]). The boundary of the singularity z0 + z1 + + zn = 0 for n odd is diffeomorphic to the standard sphere if d ≡ 1 mod 4 and to the boundary of Kervaire’s plumbing if d ≡ 3 mod 4.

Question. What happens if we allow more than one exponent ai to be ≥3?

The answer was provided by Pham, Milnor, and Brieskorn (in chronological order). We need first to define the join of two compact spaces. For information about joins we refer to Milnor’s paper [88]. Joins were a familiar construction in the old days of combinatorial topology, since the join of a simplex of dimension n with one of dimension m produces one of 9.6 Joins and Pham–Brieskorn singularities 85 dimension n + m + 1. Hence the join of two simplicial complexes is immediately a new simplicial complex. Cones and suspensions are natural objects in this setting. In order to define a join, let us begin by recalling what a mapping cylinder is. Let X and Y be (say) compact topological spaces. Let φ : X −→ Y be a continuous map. The mapping cylinder Cyl(φ) is obtained from the disjoint union of X × [0, 1] and Y by identifying (x, 1) with φ(x) ∈ Y for each x ∈ X. For example, the cone with base X and vertex the point P is the mapping cylinder of the constant map φ : X −→ {P}. There is a natural inclusion X ,→ Cyl(φ) given by x 7→ (x, 0) and Cyl(φ) canonically deformation retracts onto Y. Now let A and B be two compact spaces. Consider the product A × B. We have two projection maps πA : A × B −→ A and πB : A × B −→ B.

Deﬁnition 9.15. Let Cyl(πA) and Cyl(πB) be the two mapping cylinders. The product A × B is naturally included in both. The join A ? B is obtained by identifying point to point the two copies of A × B in the disjoint union of Cyl(πA) and Cyl(πB). The join is by construction decomposed into the two mapping cylinders. The Mayer–Vietoris exact sequence associated to this decomposition easily gives rise to the following Milnor theorem about the reduced homology of a join. Theorem 9.16. Let A and B be two compact C.W. complexes. Let A ? B be the join of A and B. Then Õ Hbm+1(A ? B) = Hbi(A) ⊗ Hbj(B) i+j=m where Hb denotes reduced homology. The joins appear to describe the topology of a germ f = g⊕h : (Cn+m, 0) → (C, 0) where g : (Cn, 0) → (C, 0) and h : (Cm, 0) → (C, 0) are germs of holomorphic functions with an isolated critical point at the origin and

f (x1,..., xn, y1,..., ym) = g(x1,..., xn) + h(y1,..., ym).

One has the following results: (1) The germ f = g ⊕ h has an isolated singularity at the origin and its Milnor fiber has the homotopy type of the join of the Milnor fiber of g with the Milnor fiber of h. (2) The monodromy of f is the tensor product of the monodromies of g and h. This is Sebastiani–Thom’s result [110]. (3) The Seifert form associated to the Milnor fiber of f is the tensor product of the Seifert form of g with the Seifert form of h. This is Koichi Sakamoto’s result [109]. 86 9 Singularities of complex hypersurfaces

(4) One can even do better. In [57] Kauffman and Neumann get an explicit description of the fiber (not only of its homotopy type) and recover the results we have just quoted. See below. In particular, the join construction allows the singularities given by the polynomials n Õ ai P(z0, z1,..., zn) = zi i=0 to be studied by induction on n. Indeed, the topology of the Milnor fiber of a germ ai zi equipped with its monodromy is very elementary: It is the cyclic permutation of ai points. Let FP be the Milnor fiber associated to P. By tensor product the action of the monodromy on H(FP, C) is easy to compute. In [15], Brieskorn determines when the link of a Pham–Brieskorn singularity is a homotopy sphere and then which element of bPn+1 it represents.

9.7 The paper by Kauﬀman [56]

Let f : (Cn, 0) → (C, 0) be a germ of holomorphic function with an isolated critical point at the origin in Cn. Let us take an integer a ≥ 2. We deﬁne a new germ n+1 fa : C , 0 −→ C, 0 by a fa(z0, z1,..., zn−1, zn) = f (z0, z1,..., zn−1) + zn .

Our objective is to understand the topology of fa from the topology of f and the integer a. For a topologist it is clear that the boundary La of fa is the cyclic covering 2n−1 of S of degree a ramified on the boundary Lf of f . In fact this idea is quite old and goes back at least to Heegaard and Wirtinger around 1900. It comes from the early attempts to understand the ramification of multivalued algebraic functions of several variables. For the same reason, it is clear that the Milnor fiber Fa of fa is the −1 2 2n cyclic covering of degree a of the crooked ball N( f ) = f (Bη) ∩ B ramified on the Milnor fiber F of f . 2n+1 But how can we describe the open book decomposition S with binding La? Kauffman proceeds like this. The Galois group of the ramified cyclic covering Fa −→ N( f ) has a natural generator (coming from the complex orientation). It could be used as a monodromy for an open book decomposition, but for a little difficulty: It is not the identity on the boundary. Then Kauffman modifies it to obtain a new monodromy. Thus we obtain an open book decomposition of S2n+1 that is isomorphic to the one associated to fa. The main part of Kauffman’s paper generalizes this situation to the one where the singularity f is replaced by an open book decomposition of S2n−1 with page a handlebody. 9.8 The paper by Kauffman and Neumann [57] 87

One very pleasant consequence of the paper is that one obtains a complete topological description of Pham–Brieskorn singularities by an iterative process that takes the shape of stairs. One has a complete vision of the Milnor ﬁber, not “only” of its homotopy type.

9.8 The paper by Kauﬀman and Neumann [57]

Let us conclude this little “survey” by coming back to [57]. The general situation considered by the authors is the following: K is a pair Kk−2 ⊂ Sk where K is a compact, oriented submanifold without boundary and L = Ll−2 ⊂ Sl is an embedding where L is the binding of an open book decomposition of Sl. There is no condition on the parity of k ≥ 1 or l ≥ 1. No assumption of connectedness is made. From this data the authors construct a new pair K ⊗ L = K ⊗ L ⊂ Sk+l+1, where K ⊗ L denotes a compact, oriented submanifold without boundary, of dimension k + l − 1. It is remarkable that the dimension l = 1 is allowed. In this case L = ∅ and the ﬁbration S1 −→ S1 can be any cyclic covering. If d is the degree of the covering, they denote by [d] the corresponding “open book decomposition”. The case k = 3 is also remarkable since it applies to classical knots and links in S3. From these, using any open book decomposition on some sphere Sl, the authors construct new links in Sl+3+1, with various properties. They have a control on the various Seifert pairings. For example, let us consider ( ) 2 2 3 f1 u0, u1 = u0 + u1. The link of f1 is a Hopf link in S , which we denote by H. We have seen (Chapter 9.4) that its Seifert matrix is (−1). If k ≥ 3 is odd, the link K ⊗ H is a link in Sk+4 with a Seifert form equal to minus the Seifert form of K, once a Seifert hypersurface is chosen for K. Then, if K is a knot (i.e., an embedded (k − 2)-homotopy sphere in Sk), K ⊗ H is an embedded (k + 2)-homotopy sphere in Sk+4. This topological construction explains the periodicity in knot cobordism discovered algebraically by Jerry Levine, for which Glen Bredon had proposed a direct topological proof without writing down the details. See Figure F.2 for a possible representation of H. It has the correct Seifert matrix.

About singularities. In Kauﬀman–Neumann’s paper, links have to be manifolds. This excludes nonisolated singularities, since the boundary of the singularity is not a manifold in this case (it is a pseudo-manifold, which is good to know, but not enough). Hence the paper applies to isolated singularities only, and for them we have an open book decomposition. Their paper is another approach to the f ⊕ g situation (see Section 9.7). 88 9 Singularities of complex hypersurfaces

9.9 Kauﬀman–Neumann’s construction when both links are the binding of an open book decomposition

The data is the following. We have a link K = K ⊂ Sk and a link L = L ⊂ Sl. Both are the binding of an open book decomposition. We denote by E(K) the exterior of K in Sk and by E(L) the exterior of L in Sl. We denote the projection of the exterior on S1 by α, resp. β, and we consider the following diagram, where both α and β are locally trivial ﬁbrations: E(L)

↓ β

α E(K) −→ S1. By taking the pullback we have

α∗ E(K) ×S1 E(L) −→ E(L)

↓ β∗ ↓ β

α E(K) −→ S1.

Lemma 9.17. The space E(K) ×S1 E(L) is a manifold of dimension (k + l − 1). Its boundary is the union of K × E(L) and E(K) × L.

Proof. By the properties of pullbacks (see Steenrod’s book [120, pp. 47–49]), the maps α∗ and β∗ are projections of locally trivial bundles. Let us denote the fiber of α (which is the same as the fiber of α∗) by F and the fiber of β (which is the same as the fiber of β∗) by G. We shall need the fibers a little bit later. α×β 1 1 1 1 Consider the product map E(K) × E(L) −→ S × S . Let ∆S1 ⊂ S × S be the diagonal. By definition of a pullback of bundles (see [120, Section 10.2, second definition]), the space E(K) ×S1 E(L) is the inverse image by α × β of the diagonal ∆S1 . Now the map α × β is transverse to the diagonal. Hence the inverse image is a manifold of dimension (k + l − 1). We now consider its boundary. The fibration of the boundary bE(K) of E(K), 1 α|bE(K) : bE(K) −→ S is trivial and we choose a trivialization to identify bE(K) 1 with K × S . Analogously the fibration of the boundary bE(L) of E(L), β|bE(L) : bE(L) −→ S1 is trivial and we choose a trivialization to identify bE(L) with S1 × L. The boundary of E(K)×S1 E(L) is therefore the union E(K)×L with K×E(L). The intersection of the two pieces is the “corner” bE(K)×L = K×S1×L = K×bE(L). 9.9 Kauffman–Neumann’s construction 89

On the other hand we consider the decompositions Sl = (D2 × L) ∪ E(L) and Sk = (K × D2) ∪ E(K). Hence we have

b(K × Dl+1) = K × Sl = (K × D2 × L) ∪ (K × E(L)) and

b(Dk+1 × L) = Sk × L = (K × D2 × L) ∪ (E(K) × L).

l+1 k+1 Then we take the disjoint union of E(K) ×S1 E(L) with K × D and with D × L. • On the boundary of K × Dl+1 we identify K × E(L) with its counterpart on the boundary of E(K) ×S1 E(L). • On the boundary of Dk+1 × L we identify E(K) × L with its counterpart on the boundary of E(K) ×S1 E(L). • Finally on the boundary of K×Dl+1 and Dk+1×L we identify the two K×D2×L. This is the new link K ⊗ L defined abstractly. The reader can check as a consolation that the dimension of each piece is equal to k + l − 1. The embedding of K ⊗ L in Sk+l+1 to get K ⊗ L is beyond the reach of this small overview, since it requires another technique, defined by the authors and called branched fibration. Here are some other nice results, still for links that are bindings of an open book decomposition. (1) The link K ⊗ L is also the binding of an open book decomposition. (2) Let J be the page (fiber) of the open book of K ⊗ L. Then the join F ?G embeds in J and the embedding is a homotopy equivalence. Moreover, the homotopy equivalence is equivariant with respect to the join of the monodromies for K and L acting on F ? G and the monodromy acting on J. (3) Let f : (Cn, 0) −→ (C, 0) and g : (Cm, 0) −→ (C, 0) be two analytic germs with isolated singularities at the origin and f ⊕ g defined in Section 9.6. Then the tensor product of the open book decompositions associated to f and g produces an open book decomposition isotopic to the one associated to f ⊕ g.

Linking numbers and signs

In the literature there is an abundance of deﬁnitions and formulas about Seifert forms and matrices. Apparently they look more or less the same, but often they diﬀer in detail, in particular in signs. However, in some situations signs may be important and are not arbitrary. This is especially the case in complex geometry, in particular in singularity theory. Therefore, in the hope of avoiding ambiguity, we make the rules we follow explicit. Basically there are two of them: the boundary of an oriented manifold and linking numbers.

A.1 The boundary of an oriented manifold

Rule A.1 (The oriented boundary of an oriented manifold). Let Mm be a compact connected manifold of dimension m, with nonempty boundary ∂M. An orientation of M produces (in fact is) a well-deﬁned generator [M, ∂M] ∈ Hm(M, ∂M; Z). By deﬁnition, the orientation induced by [M, ∂M] on ∂M is the image [∂M] of this generator by the boundary homomorphism

∂ : Hm(M, ∂M; Z) → Hm−1(∂M; Z). In the simplicial case (which is adequate for the structures we consider) the boundary homomorphism is induced by the (classical) boundary homomorphism in simplicial chain complexes deﬁned by

j=q Õ j ∂(A0, A1,..., Aj,..., Aq) = (−1) (A0, A1,..., Abj,..., Aq). j=0 See, e.g., Eilenberg–Steenrod [32, p. 88]. If Mm is a differentiable manifold, an orientation of M may also be defined as a coherent choice of equivalence classes of m-frames in each fiber of the tangent bundle. Equivalently and more conveniently, it may also be defined as the choice of an atlas such that the coordinate changes have positive determinant everywhere. In this setting, the orientation induced by [M, ∂M] on ∂M is as follows: for x ∈ ∂M choose a frame (e1,..., em) such that e1 is orthogonal to ∂M at x and is pointing to the exterior of M. Then (e2,..., em) is a frame tangent to ∂M that produces the desired 92 A Linking numbers and signs orientation at x. It is important that this rule involving frames be compatible with the rule that involves the boundary operator in homology (in fact it can be deduced from it). These rules are also compatible with Stokes formula. See Conlon [28, p. 230]. See Figure F.1. Consequence. Let Uu and V v be two compact, connected and oriented manifolds, such that ∂U , ∅ and ∂V = ∅. If we wish to take their product in the oriented category, we should use the product U ×V rather than V ×U. In this way, the oriented boundary ∂(U × V) is naturally equal to (∂U) × V. For instance, if U = I = [−1, +1] it is recommendable to use I × V in that order, although many topologists consider the product in the reverse order.

A.2 Linking numbers

Rule A.2 (Linking numbers (Lefschetz)). Let γ1 be a (k−1)-cycle and γ2 be an (l−1)- m−1 m cycle in S . Suppose that γ1 ∩ γ2 = ∅ and that m = k + l. Suppose that B is m−1 m oriented; hence S = ∂B is oriented too. Then the linking number LSm−1 (γ1; γ2) m−1 of γ1 and γ2 in S is by definition the intersection number IBm (c1, c2) where ci is m a chain in B such that ∂ci = γi for i = 1, 2. We recall the usual definition of the intersection number. Without loss of generality we may assume that the intersection points of c1 and c2 are finite in number and that the intersection is transversal at such a point P. At P we choose a frame representing the orientation of c1 and a frame for c2. We place side by side these two frames and compare the orientation of Bm at P thus obtained with the given one. The sum of ±1 we obtain is the intersection number we are looking for. It follows immediately that kl IBm (c1, c2) = (−1) IBm (c2, c1). Hence the same symmetry rule is valid for the linking number:

kl LSm−1 (γ1; γ2) = (−1) LSm−1 (γ2; γ1).

A consequence of Rules A.1 and A.2 is the following.

m−1 Consequence. Let γ1 and γ2 be cycles in S as in the statement of Rule 2. Let βi m−1 be a chain in S such that ∂βi = γi for i = 1, 2. Then

k LSm−1 (γ1; γ2) = ISm−1 (γ1, β2) = (−1) ISm−1 (β1, γ2).

See Figures F.2 and F.3. B

Existence of Seifert hypersurfaces

Let Mn+2 be a closed, oriented, 2-connected, (n + 2)-dimensional differentiable manifold, and let Ln be a closed, oriented, n-dimensional differentiable submanifold of Mn+2. (We do not assume that Ln is connected.) Definition B.1. A Seifert hypersurface of Ln is a differentiable, oriented and connected, (n + 1)-dimensional submanifold Fn+1 in Mn+2 that has Ln as boundary. The aim of this chapter is to prove the following theorem: Theorem B.2. A submanifold Ln ⊂ Mn+2 has Seifert hypersurfaces. Lemma B.3. The normal bundle of Ln is trivial. Proof. The normal bundle of Ln is orientable. Since it has rank 2, it is enough to prove that it has a nowhere-vanishing section. The only obstruction to construct such a section is the Euler class e ∈ H2(Ln, Z). But H2(Mn+2, Z) = 0 since Mn+2 is 2-connected. Hence Thom’s formula for the Euler class implies that e = 0. Let N(L) be a compact tubular neighborhood of Ln. Let ν : N(L) → Ln be the projection of the normal disc bundle associated to N(L). Let bN(L) be the boundary of N(L) and E(L) = Mn+2 \ N(˚L) be the exterior of Ln. n Definition B.4. For each connected component Li of L , 1 ≤ i ≤ r, we choose −1 xi ∈ Li.A meridian mi of Li is a circle mi = bN(L) ∩ ν (xi) oriented as the −1 n 2 n boundary of ν (xi). Let π : (L × B ) → L be the projection on the first factor. A trivialization of ν is a diffeomorphism φ : N(L) → (Ln × B2) such that ν = π ◦ φ.

Remark. The homology group Hn(N(L), Z) is free with a basis given by the fundamental classes li of Li in N(L). By Alexander duality, H1(E(L), Z) is free with a basis given by the homology classes of the meridians that we will also denote by mi. Proof of Theorem B.2. We prove that there exists a differentiable map ψ : E(L) → S1 such that the restriction ψ0 of ψ on bN(L) extends to a fibration ψ˜ : N(L) → B2 where Ln = ψ˜ −1(0). Then, we choose a regular value z of ψ and we denote by I 2 n+1 ( −1( )) ∪ ( ˜ −1( )) n the radius in B of extremity z. By construction F1 = ψ z ψ I has L n+1 as boundary. If F1 is not connected, first we remove its closed components, then n+1 we connect the connected components of F1 with the help of carefully chosen thin tubes, and we obtain a connected Fn+1. 94 B Existence of Seifert hypersurfaces

Differentiable maps ψ : E(L) → S1 are classified up to homotopy by H1(E(L), Z). The universal coefficient theorem and the above remark imply that H1(E(L), Z) is ∗ free and has a basis given by the duals mi , 1 ≤ i ≤ r of the meridians. We choose a 1 differential map ψ : E(L) → S , which has degree +1 on each mi , 1 ≤ i ≤ r. On the other hand, let φ be a trivialization, φ : N(L) → Ln × B2, of the normal bundle of Ln. Let π0 : Ln × B2 → B2 be the projection on the second factor. The 0 0 restriction φ of π ◦ φ on bN(L) has degree +1 on each meridian mi, 1 ≤ i ≤ r. We will say that such a map φ0 : bN(L) → S1 is given by the parametrization φ. As H1(Mn+2, Z) = 0 and H2(Mn+2, Z) = 0, we have

1 1 1 H (bN(L), Z) = iN (H (N(L), Z)) ⊕ iE (H (E(L), Z)) where iN and iE are respectively induced by the inclusions of bN(L) in N(L) and E(L). So, a differentiable map α : bN(L) → S1 is homotopic to a map given by a parametrization if and only if α has degree +1 on each meridian mi, 1 ≤ i ≤ r. Let ψ0 be the restriction on bN(L) of the already chosen map ψ : E(L) → S1. Hence ψ0 is homotopic to a map φ0 given by a parametrization φ. If necessary, we perform this homotopy in a small collar around bN(L) and we can extend ψ0 to the fibration π0 ◦ φ : N(L) → B2. Proposition B.5. If Fn+1 is a Seifert hypersurface of Ln, there exists a differentiable map ψ : E(L) → S1 such that ψ is of degree +1 on a meridian m and F = Fn+1 ∩ E(L) = (ψ−1(z)), where z is a regular value of ψ. Proof. As F = Fn+1 ∩ E(L) is oriented, F has a trivial normal bundle. Hence, we can choose an embedding α : I × F → E(K) where I = [−1, +1], such that the image α(I × F) = N(F) is an oriented compact tubular neighborhood of F in Sn+2 and F = α(0 × F). We use the following notation: F+ = α(+1 × F) and F− = α(−1 × F). Let E(F) = (E(K)\ α(F× ] − 1, +1[ )) be the exterior of F. We can define ψ as follows: when y = α(x, t) ∈ N(F), let ψ(y) = eiπt ; when y ∈ E(F) , let ψ(y) = −1. C

Open book decompositions

C.1 Open books

Open books were introduced (without the name) by John Milnor in [93]. We use the following definition: Definition C.1. Let Mm be a closed, oriented, m-dimensional differentiable manifold, and let Lm−2 be a closed, oriented, (m − 2)-dimensional differentiable map submanifold of Mm. Let N(L) be a compact tubular neighborhood of Lm−2 and E(L) = Mm \ N˚(L) be the exterior of Lm−2. An open book decomposition of Mm with binding Lm−2 is given by a differentiable fibration ψ over the circle:

ψ : E(L) → S1, such that the restriction ψ0 of ψ on bN(L) extends to a fibration ψ˜ : N(L) → B2 and Lm−2 = ψ˜ −1(0). A fiber F of ψ is a page of the open book decomposition. The fibration ψ˜ produces an isotopy, in N(L), between Lm−2 and the boundary of F. Remarks. (1) When we have an open book decomposition of Mm with binding Lm−2, the fibration ψ˜ defined on N(L) is trivial and the normal bundle of Lm−2 is trivial. Hence there exists a trivialization of N(L) such that the restriction of the fibration ψ : E(L) → S1 to the boundary bE(L) coincides with the projection of L × S1 on the second factor. (2) Let F be a fiber of ψ and let h : F → F be a monodromy of ψ. The exterior E(L) is diffeomorphic to the mapping torus T(F; h) of the monodromy h. Recall that it is the quotient of F × [0, 1] by the gluing (x, 1) ∼ (h(x), 0) for x ∈ F. Point (1) implies that we can choose h such that its restriction on bF is the identity. Theorem C.2 (Uniqueness theorem for open book decompositions of Sn+2). Let Ln ⊂ n+2 n+2 S be a compact, simply connected submanifold in S such that π1(E(L)) = Z. Suppose that n ≥ 4. Then two open book decompositions of Sn+2 with binding Ln are isomorphic. It is not difficult to replace Sn+2 by more general manifolds. 96 C Open book decompositions

Since Ln is simply connected, there is a unique trivialization of its normal bundle. Hence the main point to prove the uniqueness is to prove the uniqueness of the fibration over the circle. Explicitly, we are given two fibrations f and ef : E(L) → S1. We wish to prove that they are isomorphic. Proof. The proof proceeds in two steps: (1) By Browder’s lemma 2 (see [17]), there exists a diffeomorphism Φ : E(L) → E(L) such that Φ( f −1(1)) = ef −1(1). In other words, Φ sends one fiber of f onto one fiber of ef . (2) By the Cerf pseudo-isotopy theorem (see [27]), one can modify Φ to a diffeomorphism Ψ which is an isomorphism between the fibrations. For details and explicit formulas, see Mitsuyoshi Kato [55, p. 461].

C.2 Browder’s lemma 2

Lemma C.3 (Browder’s lemma 2 in [17]). Let W be a closed, connected differentiable manifold of dimension ≥6. Let f and ef be two fibrations W → S1. We suppose that (1) f and ef are homotopic; (2) the fibers f −1(1) (resp. ef −1(1)) of f (resp. ef ) are simply connected. Then there exists a diffeomorphism h : W → W, pseudo-isotopic to the identity, such that − − h( f 1(1)) = ef 1(1). Remark. If we consider disjoint lifts of the fibers f −1(1) and ef −1(1) in the infinite cyclic covering of W, the region between them is an h-cobordism. Hence the two fibers are diffeomorphic. But we need a diffeomorphism of W that sends one to the other. The main ingredient in Browder’s proof of the lemma is Browder–Levine’s existence theorem for fibrations over the circle in a relative form and Smale’s h-cobordism theorem. There exists a version of the lemma if ∂W , ∅. The proof is essentially the same as the one for the empty boundary case. It is this version that we use. Additional conditions are (3) the boundary ∂W of W is a product Z × S1; C.2 Browder’s lemma 2 97

(4) the projections f and ef coincide on the boundary ∂W and are equal to the projection Z × S1 → S1 on the second factor. In our use of the lemma, assumption (1) is satisfied since both projections represent the same generator of H1(E(L); Z), taking orientations into account. Assumption (2) is satisfied since we require that π1E(L) = Z. Assumptions (3) and (4) are satisfied by the open book conditions.

Handlebodies and plumbings

D.1 Bouquets of spheres and handlebodies

Handlebodies are the simplest of manifolds with boundary. They were studied by Smale and Wall in the early 1960s. See [117] and [128, 129, 131]. We present here the basic facts on parallelizable handlebodies of type (q, r).

q q 2q Deﬁnition D.1. Let ϕj : (bBj )× Bj → bB , 1 ≤ j ≤ r be r disjoint embeddings. A handle presentation of type (q, r) is a manifold W2q obtained from the disjoint union 2q q q q q 2q B qj Bj × Bj by identifying x ∈ (bBj ) × Bj with ϕj(x) ∈ bB .A q-handlebody is a manifold diﬀeomorphic to a handle presentation of type (q, r).

See Figures F.11 and F.12. q q 2q The image of Bj × Bj in W is called a handle of index q and the image Cj of q Bj × {0} is its core. The collection of embeddings Φ = {ϕj }j is the attaching map q−1 of the presentation and L = qj ϕj(Sj × {0}) is its attaching link. If q , 2 each connected component Lj of the link L of the handle presentation bounds an oriented 2q q-ball Dj in B . Let cj be the q-cycle obtained as the union of Dj with the core Cj. We say that {cj, 1 ≤ j ≤ k} is an adapted (to the handle presentation) basis of 2q Hq(W , Z). Inspired by Smale, we denote by H(q, r) the set of diffeomorphism classes of manifolds that admit a presentation as above. One can say equivalently that such manifolds admit a Morse function with 1 minimum and r critical points of index q. This is true without any restriction on the integers (q, r). The case q = 1 is easily understood, often with arguments different from those presented here. It is omitted from our discussion. When q ≥ 2, a q-handlebody is orientable. We assume that it is oriented. Clearly a q-handlebody has the homotopy type of a bouquet of k spheres of dimension q. The converse is trivially true if q = 1 but wrong if q ≥ 2. Counterexamples are, e.g., provided by homology spheres of dimension (2q −1), with nontrivial fundamental group, which bound a contractible 2q-manifold. This contractible manifold cannot be a q-handlebody since it would be a 2q-ball and hence its boundary would be S2q−1. Differential homology spheres that bound differential contractible manifolds are plentiful. Indeed, Kervaire proved in [65] the following theorem. 100 D Handlebodies and plumbings

Theorem D.2. Let Mn be a diﬀerential, oriented homology sphere with n , 3. Then there exists a homotopy sphere Σn such that the connected sum Mn]Σn bounds a diﬀerential contractible manifold.

Remarks. (1) Kervaire proves also that Σn is unique.

(2) To take the connected sum with a homotopy sphere amounts to changing the diﬀerential structure of Mn in the neighborhood of a point.

(3) Rohlin’s theorem prevents the theorem being true in the diﬀerential category if n = 3. However Michael Freedman proved that every homology 3-sphere bounds a topological contractible 4-manifold. See [36].

Theorem D.3 (Recognition theorem). Suppose that q ≥ 3. LetV be a 2q-dimensional manifold that has the homotopy type of a bouquet of spheres of dimension q. Then V is a q-handlebody if and only if bV is simply connected.

red Remark. An easy computation reveals that the reduced homology groups Hi (bV; Z) vanish for i ≤ q − 2. Hence, if q ≥ 3, the boundary bV is (q − 2)-connected if and only if it is simply connected.

Proof of Theorem D.3. By general position, the boundary of a q-handlebody is simply connected if q ≥ 3. Conversely, assume that bV is simply connected and let us prove that V is a handlebody. The original proof is due to Smale. Here is a slightly different one that makes the simple connectivity of the boundary quite visible. This proof rests on a classical engulfing argument and on Smale’s h-cobordism theorem. Choose a basis {ej } for j = 1,..., r of Hq(V; Z) = πq(V). Since q ≥ 3 and since q π1(V) = 1 we can represent each ej by an embedded sphere Sj ,→ V (by Whitney). q q By general position, we can assume that Si ∩ Sj is a finite set of points. In each q sphere Sj we construct an embedded arc γj that contains all the intersection points q q of Sj with the other spheres Si for i , j. The union of the arcs γj for j = 1,..., r is a compact graph Γ embedded in V. Let CΓ be the abstract cone with base Γ. Since V is simply connected the inclusion Γ ⊂ V extends to a map η : CΓ → V. Since dim(V) ≥ 6 we can find an η which is a piecewise regular embedding (in the sense q of Morris Hirsch) such that η(CΓ) ∩ Sj = γj for j = 1,..., r. Let N be regular neighborhood of η(CΓ) in V. By the Whitehead–Hirsch theory of regular neighborhoods in the differential category, N is a 2q-ball B2q in V that q q meets each sphere Sj in a regular neighborhood of γj, hence in a q-ball Dj . Let q q ˚ q q q q Bj = Sj \ Dj . By construction we have Bi ∩ Bj = ∅ for i , j. Hence the Bj are the 2q cores of handles of index q attached to the ball B . Let hj be the handle with core q 2q Bj obtained by thickening the core a little bit. The union B ∪j hj is a q-handlebody W embedded in the interior of V. D.2 Parallelizable handlebodies 101

Consider X = V \ W˚ . We claim that X is a simply connected h-cobordism. By construction, the inclusion W ⊂ V induces an isomorphism on homology. Hence by excision H∗(X, bW; Z) = 0. By Poincaré duality, H∗(X, bV) = 0. The boundary bW is simply connected, since W is a q-handlebody with q ≥ 3. Now V = W ∪ X with W ∩ X = bW. The manifolds V, W, and bW are simply connected. Hence by van Kampen, X is also simply connected. To conclude that X is a simply connected h-cobordism we still need that bV is simply connected. It is here that the hypothesis is used. By Smale, X is a product since dim X ≥ 6.

Corollary D.4 (Of the proof of the theorem). If q ≥ 3, any basis of Hq(V; Z) can be realized by the cores of a handle decomposition.

Now we wish to characterize the attaching map Φ = {ϕj }j up to isotopy. We assume that q ≥ 3. q q−1 Each ϕj((bBj ) × {0}) is an oriented sphere Kj of dimension (q − 1) embedded in the sphere bB2q of dimension (2q − 1). This knot is trivial and the isotopy class of the oriented link K = {Kj }j is completely determined by the linking numbers LbB2q (Ki, Kj) ∈ Z for i , j. See the section below for more details. q−1 q 2q The sphere Kj bounds a diﬀerential ball Dj in B . The normal bundle of this ball has a unique trivialization, which by restriction provides a canonical trivialization 2q−1 2q of the normal bundle of Kj ⊂ bB . The attaching map ϕj provides another trivialization of this normal bundle. The comparison between the two trivializations produces an element Qj ∈ πq−1(SOq). The next proposition summarizes what we have obtained so far.

Proposition D.5. Up to isotopy the attaching map Φ is characterized by the linking co- eﬃcients LbB2q (Ki, Kj) for i , j and the set of elements {Qj }j with Qj ∈ πq−1(SOq).

The proof of the proposition follows easily from Haeﬂiger’s classiﬁcation of spherical links presented in Section D.3.

Comment. The homotopy groups πq−1(SOq) are not stable. Their value is also periodic of period 8 as for the stable ones. See [131]. We shall not need their value here.

D.2 Parallelizable handlebodies

We now consider parallelizable (equivalently, stably parallelizable) q-handlebodies. This is what we really need from the theory of handlebodies since Seifert hypersurfaces are parallelizable. 102 D Handlebodies and plumbings

Consider a part of the exact homotopy sequence of the locally trivial ﬁbration q SOq+1 → S with ﬁber SOq:

q ∂ σ · · · −→ πq(S ) −→ πq−1(SOq) −→ πq−1(SOq+1) = πq−1(SO) −→ · · · .

Comments. (1) The group πj−1(SOk) classiﬁes oriented real vector bundles (with ﬁber Rk) over the sphere S j. q q q (2) The generator of πq(S ) is the identity map i : S → S ; its image ∂(i) ∈ q πq−1(SOq) represents the tangent bundle of S (this is due to Steenrod [120, Section 23]). (3) σ is the stabilization homomorphism.

Therefore Ker(σ) = Im(∂) ⊂ πq−1(SOq) classiﬁes stably trivial q-vector bundles over Sq. This subgroup takes the following values (most of this is also due to Steenrod, together with Kervaire’s and Bott–Milnor’s nonparallelizability of spheres of dimension , 1, 3, 7; see Kosinski’s book [72, p. 231]). Theorem D.6. (i) When q is even, Ker(σ) = Im(∂) is isomorphic to the integers Z; its elements are distinguished by the Euler number of the vector bundle, which can take any even value (since the Euler characteristic of an even-dimensional sphere is equal to 2). (ii) When q is odd, there are two possibilities: • If q = 1, 3, 7, then Ker(σ) = Im(∂) vanishes. • If q , 1, 3, 7, then Ker(σ) = Im(∂) is isomorphic to Z/2. The nontrivial element is represented by the tangent bundle to Sq.

Let W be a parallelizable q-handlebody with a basis {cj, 1 ≤ j ≤ k} of Hq(W, Z) which is adapted to a handle presentation. As explained above, each q-cycle cj provides an element Qj ∈ πq−1(SOq). Since W is parallelizable, we have Qj ∈ Ker(σ) = Im(∂) ⊂ πq−1(SOq). Hence Qj ∈ Z if q is even, Qj = 0 if q = 1, 3, 7, and Qj ∈ Z/2 if q is odd and q , 1, 3, 7. One can prove that QW (cj) = Qj extends to a quadratic form

QW : Hq(W; Z) → Z if q is even;

QW : Hq(W; Z/2) → Z/2 if q is odd and , 1, 3, 7. See [72, pp. 206–208]. The next proposition is an immediate consequence of what has just been said. Proposition D.7. (1) Suppose that W is parallelizable and let q be even. Then the intersection form IW is even and one has the equality 2Q(x) = IW (x, x) for all x ∈ Hq(W, Z). D.3 m-dimensional spherical links in S2m+1 103

(2) Suppose that W is parallelizable and let q be odd and , 1, 3, 7. Then the map QW : Hq(W; Z/2) → Z/2 is a quadratic form with associated bilinear form the reduction mod 2 of the antisymmetric intersection form IW . In other words we have in Z/2 the equality QW (x + y) = QW (x) + QW (y) + IW (x, y). If this equality is satisfied, one says that I is associated to Q. Theorem D.8. Let q ≥ 3. There is a natural bijection between oriented diffeomorphism classes of oriented parallelizable q-handlebodies and (i) if q is even, isometry classes of integer-valued, symmetric, even bilinear forms over free Z-modules of finite rank; (ii) if q is equal to 3 or 7, isometry classes of antisymmetric bilinear forms over Z-modules of finite rank; (iii) if q is odd and , 3, 7, free Z-modules of finite rank equipped with an integer- valued antisymmetric bilinear form together with a quadratic form with values in Z/2 (associated as in the proposition above). The theorem is a consequence of what we have done so far and from the fact that the intersection determines the linking coefficients (use the corollary of the recognition theorem). It is a special case of Wall’s vast study of handlebodies.

Deﬁnition D.9. A q-handlebody W is unimodular if the intersection form IW is unimodular; in other words if the discriminant of IW is equal to ±1. Comments. (1) If q ≥ 3, unimodularity is equivalent to requiring that the boundary bW be a homotopy sphere. (2) If q = 2, unimodularity is equivalent to bW being a homology sphere.

D.3 m-dimensional spherical links in S2m+1

Definition D.10. An m-dimensional spherical r-link in S2m+1 is a differentiable m ⊂ 2m+1 { m m} m submanifold L S , with r connected components L1 ,..., Lr . Each Li is diffeomorphic to the standard sphere Sm and oriented. Note that the components are labeled.

Theorem D.11 (Haefliger [42]). Let m ≥ 2. Let Lm ⊂ S2m+1 and bLm ⊂ S2m+1 be two spherical r-links. There exists a diffeomorphism Φ : S2m+1 → S2m+1 such that (i) Φ is isotopic to the identity; m m (ii) Φ(Li ) = bLi for 1 ≤ i ≤ r preserving both orientations; 104 D Handlebodies and plumbings if and only if condition (♦) is satisfied:

m m m m Lk(Li ; Lj ) = Lk(bLi ; bLj ) for 1 ≤ i < j ≤ r. (♦)

Comments. (1) Note that the diffeomorphism Φ preserves all orientations and the labels. Compare with Haefliger’s definitions in [42]. (2) Condition ♦ is clearly necessary for the existence of Φ for all m ≥ 1. The point is that the condition is sufficient if m ≥ 2.

(3) Let {aij ∈ Z} be a set of integers with 1 ≤ i < j ≤ r. Then it is easy to prove m 2m+1 m m that there exists a spherical r-link L ⊂ S such that aij = Lk(Li ; Lj ). The (−1)m+1-symmetry of linking coefficients enables us to state the following corollary. Corollary D.12 (Classification). Fix m ≥ 2. There exists a natural bijection between (i) the set of isotopy classes of m-dimensional spherical r-links in S2m+1 and (ii) the set of (−1)m+1-symmetric (r ×r) matrices with integer coefficients and zeros on the diagonal. Implicit in the statement of the theorem is the fact that the individual components of the link are trivial knots.

D.4 Plumbing

The name “plumbing” was introduced by David Mumford in 1961 in his epoch- making paper [97]. There he showed (among several other things) that the topology of a resolution of a normal singularity of a complex surface can be described by a construction called by him a “plumbing fixture”. Now, Hirzebruch had fully described in his 1950 thesis [51] how to resolve these singularities, thus giving rise to what is called today the Hirzebruch–Jung resolution. Hirzebruch was impressed by Mumford’s results and he popularized the name and use of plumbing. A plumbing fixture is a way to construct some differential manifolds by gluing together “building blocks”. In the case of interest to us (to construct a parallelizable differentiable manifold with boundary), the building blocks are stably trivial q-disc bundles over a sphere of dimension q. We orient the base and the total space of the bundle. We code a building block by a weighted vertex as follows (the weight determines the stably trivial q-disc bundles up to isomorphism): D.4 Plumbing 105

(1) We attach the Euler number of the disc bundle when q is even; this is an integer that can take any even value.

(2) When q is odd and equal to 1, 3, or 7 we attach the number 0.

(3) When q is odd and , 1, 3, or 7 we attach the number 0 if the bundle is trivial or the number 1 if the bundle is isomorphic to the tangent bundle of Sq.

See Figure F.10. The plumbing construction proceeds as follows. We consider two building blocks and index them by i = 1 or 2. For each of them we choose an embedding of the q q q q q closed ball B in Si (with image Di ) and a trivialization Di × ∆i of the bundle over q q q Di (where ∆ denotes a copy of B ). The embeddings and the trivializations are assumed to be compatible with the orientations. q × ∆q q × ∆q Then we glue them together by identifying D1 1 with D2 2 with a diffeo- q ∆q morphism that is the product of two diffeomorphisms, one between D1 and 2 and ∆q q the other between 1 and D2 , both preserving the orientations. When q is even the gluing diffeomorphism is orientation preserving while it is orientation reversing if q is odd. Hence in the latter case we have to reverse the orientation of one of the two blocks to get an oriented manifold. The result is a parallelizable 2q-manifold with corners. This is an elementary positive plumbing. When the two diffeomorphisms, one q ∆q ∆q q between D1 and 2 and the other between 1 and D2 , are both reversing the orientations, this is an elementary negative plumbing. Comments. For the resolution of surface singularities (q = 2) treated by Hirzebruch or Mumford, the blocks are disc bundles over any oriented closed (real) surfaces, in general not parallelizable. The boundaries of such plumbings have been classified by Neumann in [99]. We can construct more complicated manifolds by starting with several blocks and plumbing them more or less at will. Here is a standard way to represent some plumbing data by a graph (when q is odd, we will consider only plumbings represented by trees in order to avoid the orientation problem mentioned above). To each vertex we attach a weight, which represents the weight of a block. Each edge represents an elementary positive plumbing. We perform the plumbings according to the given data. We obtain an oriented, parallelizable 2q-dimensional manifold with corners. The tradition is to not bother about corners, since they can be smoothed. The result is well defined up to orientation-preserving diffeomorphism. The most popular trees are

(i) for q odd and , 1, 3, 7, the tree denoted usually by Ad in the theory of Cartan algebras, but weighted with 1 at the vertices, where the index d is the number 106 D Handlebodies and plumbings

of vertices; e.g, the plumbing graph A5:

1 1 1 1 1 u u u u u;

(ii) for q even, the famous diagram E8 with weights +2: 2 u 2 u

22 2 2 2 2 u u u u u u.

From now on, we focus on the case q even. In order to gain full generality we need to introduce two more tools:

(1) Plumbing data coded by a ﬁnite graph Γ without loops at the vertices: The edges have a weight ± that represents the sign of the plumbing.

(2) Surgery on the boundary, which amounts to attaching 2-handles on the boundary: This is compulsory if we wish to obtain a manifold that is simply connected, since step (1) produces a non-simply connected manifold when the graph Γ has cycles.

All this is explained in detail in William Browder’s book [19, Chapter V, Section 2].

Summary. We consider oriented, parallelizable, diﬀerentiable manifolds of dimension 2q (q even), up to orientation-preserving diﬀeomorphisms. With these elements at hand we construct four sets as follows.

(1) The elements of set W1 are those manifolds that are (q − 1)-connected with (q − 2)-connected boundary.

(2) The elements of set W2 are those manifolds that have the homotopy type of a wedge of q-spheres and have a simply connected boundary.

(3) The elements of set W3 are parallelizable q-handlebodies.

(4) The elements of set W4 are those manifolds that are obtained by plumbing as explained above, including negative plumbings, plumbing data coded by a graph, and 2-handles attached on the boundary. D.4 Plumbing 107

Theorem D.13. If q ≥ 4, there is a natural bijection between any two of the sets W1, W2, W3, W4. Sketch of proof. First we suppose that q is even. We introduce the ﬁfth set S of isometry classes of symmetric, even bilinear forms on free, ﬁnite-dimensional Z- modules. For each set Wi (i = 1, 2, 3, 4) there is a map Ii : Wi −→ S that associates to each manifold its intersection form. One proves that each Ii is a bijection (see Browder’s book [19, Chapter V, Section 2]).

Note on orientations and signs. Let π : E −→ Sq be an orientable q-disc bundle on the sphere Sq. If we choose an orientation b on the base and f on the fiber F, we define then an orientation e on E by the rule e = (b, f ) meaning b first and then f . Note that (b, f ) = ( f, b) if q is even while (b, f ) = −( f, b) if q is odd. Let Φ = (φ1, φ2) be the identification diffeomorphism used for the plumbing, with q −→ ∆q ∆q −→ q diffeomorphisms φ1 : D1 2 and φ2 : 1 D2 . We denote its action on the orientations by ϕ. For a positive plumbing we have ϕ(b, f ) = ( f, b) = (b, f ) if q is even; hence Φ preserves the orientations. But we have ϕ(b, f ) = ( f, b) = −(b, f ) if q is odd; hence Φ reverses the orientations. In the latter case, to have an orientation on the manifold obtained by plumbing we reverse the order (base, fiber) in the second factor. Note that this depends on the numbering of the factors. For a negative plumbing, we have ϕ(b, f ) = (− f, −b) = ( f, b). The conclusion is as above. q q In order to compute the intersection between S1 and S2 in the plumbed manifold we proceed as follows: q · q q · q • For a positive plumbing we have S1 S2 = S1 F1 = +1 in the first factor and q · q q · q we have S1 S2 = F2 S2 = +1 in the second factor. This equality is true whatever the parity of q is, thanks to the change of orientation for odd q. The computation shows also that the result does not depend on the factor used to perform the computation. q · q q ·(− q) − • For a negative plumbing we have S1 S2 = S1 F1 = 1 for the computation q · q (− q · q) − with the first factor and S1 S2 = F2 S2 = 1 if we use the second factor q · q to perform the computation. Hence the intersection number S1 S2 is negative.

Summary. A positive or a negative plumbing preserves the orientations if q is even and reverses them if q is odd. The intersection number between the spheres is equal to +1 for a positive plumbing and it is equal to −1 for a negative plumbing.

Homotopy spheres embedded in codimension 2 and the Kervaire–Arf–Robertello–Levine invariant

E.1 Which homotopy spheres can be embedded in codimension 2?

Let Σn be a homotopy sphere differentiably embedded in Sn+2. To answer the question of which homotopy spheres can be embedded in codimension 2, let us first discuss the case n ≤ 4: • If n = 1 or 2, there is no problem since Σn = Sn. • If n = 3, we could invoke Perelman. But there is no need to do that, since Wall proved the lovely result that any 3-manifold embeds differentiably in S5. • If n = 4, we have to argue a little bit. By Kervaire–Milnor’s [67, Theorem 6.6] we know that every homotopy 4-sphere Σ4 bounds a contractible 5-manifold ∆5. Let us take the double of ∆5, i.e., two copies of ∆5 glued along their boundary. We thus get a homotopy 5-sphere Σ5. But Kervaire–Milnor proved that any homotopy 5-sphere is diffeomorphic to S5 (see [67, p. 504]). Therefore Σ4 embeds in S5 and a fortiori in S6. Note. The arguments for n = 3 and 4 are those of Kervaire in his paper about the group of a knot. Now we consider the set Gn of orientation-preserving diffeomorphism classes of oriented homotopy spheres Σn that can be embedded in Sn+2. Theorem E.1. Suppose that n ≥ 5. n (1) Then G is a finite cyclic group, canonically isomorphic to Z/dn. n n (2) If n is even, dn = 1, or in other words Σ = S . (3) Suppose that n is odd, equal to 4k − 1, with k ≥ 2. Then 4Bk d = 22k−2(22k−1 − 1) numerator . n k

The canonical generator 1 ∈ Z/dn is represented by the boundary of the E8 plumbing. 110 E Homotopy spheres and the KARL invariant

(4) If n = 4k + 1 with k ≥ 1, the group Gn is either trivial or isomorphic to Z/2. More precisely, n (41) G is trivial if n = 5, 13, 29, 61, and possibly 125 (these are the exceptional integers); n (42) G is isomorphic to Z/2 if n is not exceptional. The nontrivial element is represented by the boundary of the Kervaire plumbing.

The integer dn when n = 4k − 1 was called by Kervaire in a letter to Haefliger “l’entier bordélique bien connu”. The numerator of Bernoulli numbers is indeed a tough object, contrary to the denominator. Items (1), (2), and (3) are an immediate consequence of Kervaire–Milnor, up to a factor 2, which was settled by the solution of the Adams conjecture. Item (4) is intimately related to the Kervaire invariant problem solved in 2009. The next proposition is the first step towards the proof of the theorem above. Proposition E.2. A homotopy n-sphere can be embedded differentiably in Sn+2 if and only if it bounds a parallelizable (n + 1)-manifold. Sketch of proof. If Σn is embedded in Sn+2 it bounds a Seifert hypersurface, which is parallelizable since it is orientable and embedded in codimension 1. (Of course this argument works with no restriction on n.) Conversely if Σn bounds abstractly a parallelizable manifold, it also bounds a parallelizable handlebody. Since n ≥ 5, this is true by easy surgery below the middle dimension and by the recognition theorem (it is here that (n+1) ≥ 6 is required). It is then easy to embed such handlebodies in codimension 1. This implies the following proposition. Proposition E.3. Let n ≥ 5. The group Gn is isomorphic to Kervaire–Milnor’s group bPn+1. Question. If Σ2q−1 ⊂ S2q+1 is an odd-dimensional knot, can we detect by knot invariants which homotopy sphere is represented by Σ2q−1? The answer is yes, and the Seifert form does the work. There are two cases: (1) If n = 4k − 1 = 2q − 1 (i.e., q is even), the answer is straightforward. Take the Seifert form of any Seifert hypersurface for Σn ⊂ Sn+2, symmetrize it, take the signature of this bilinear symmetric form (which is the intersection form of the chosen Seifert hypersurface), and divide it by 8. The reduction modulo dn of the n integer so obtained is the element of Z/dn represented by Σ . (2) If n = 4k + 1 = 2q − 1 (i.e., q is odd), the answer is provided by the Kervaire– Arf–Robertello–Levine invariant as we shall see now. E.2 The Kervaire–Arf–Robertello–Levine invariant 111

E.2 The Kervaire–Arf–Robertello–Levine invariant

KARL stands for Kervaire, Arf, Robertello, and Levine. They are the main contrib- utors. The origin of the construction of the KARL invariant goes back to the thesis presented by Kervaire in Paris in June 1964, precisely with the purpose of answering the question raised just above. See [64, p. 236]. The study of the invariant was then developed by Robertello in his NYU thesis (Kervaire was the director), also in 1964. See [108]. Soon afterwards, Levine simplified in [79] Robertello’s presentation of the invariant defined via Seifert forms. In fact, in the case of classical knots Robertello gave two equivalent definitions. One of them can easily apply to higher-dimensional knots. The data is the following. We have n = 4k + 1 and K4k+1 ⊂ S4k+3 is a (4k + 1)-dimensional knot, where K4k+1 is diffeomorphic to the homotopy sphere Σ4k+1. The case n = 1 (i.e., k = 0) is admitted. Let F4k+2 be a Seifert hypersurface for K4k+1. We denote by H the quotient of H2k+1(F; Z) by its Z-torsion subgroup. Let A : H × H → Z be the Seifert form defined by A(x, y) = Lk(x; i+ y).

Deﬁnition E.4. We deﬁne the quadratic form QF : H ⊗ Z/2 → Z/2 by QF (x) = Lk2(x; i+(x)), where Lk2 denotes the linking number reduced mod 2.

Lemma E.5. We have the equality QF (x + y) = QF (x) + QF (y) + x · y where x · y denotes the intersection number in F reduced mod 2.

We denote by Arf(QF ) the Arf invariant of the quadratic form QF . Note that the bilinear form (x, y) 7→ x · y is unimodular, since ∂F is a homotopy sphere.

Theorem E.6 (Robertello; Levine [79]). Arf(QF ) is a knot invariant, i.e., it does not depend on the choice of a Seifert hypersurface. Sketch of Levine’s proof. Let p : E(K) → E(K) be the inﬁnite cyclic covering of 4k+3 −1 the knot complement E(K) = S \ N(K). Let ∆K (t) ∈ Z[t, t ] be the Alexander −1 polynomial of the knot module H2k+1(E(K); Q) normalized à la Conway by ∆(t ) = ∆(t) and ∆(1) = 1. Then Levine proves that ∆(−1) ≡ 1+4 Arf(QF ) mod 8. (Levine’s proof is a beauty.)

Deﬁnition E.7. We deﬁne the KARL invariant KARL(K) of the knot K to be Arf(QF ) for any Seifert hypersurface F. Theorem E.8 (Robertello; Levine). KARL(K) is a knot cobordism invariant. Sketch of proof, following Levine. By Levine [80], the Fox–Milnor result is true in higher dimensions. More precisely, if K is cobordant to zero, there exists an element P(t) ∈ Z[t, t−1] such that ∆(t) = P(t)P(1/t). Since ∆(1) = 1 we deduce that P(−1) is odd and hence ∆(−1) is an odd square and therefore congruent to 1 mod 8. Hence KARL(K) = 0 if K is cobordant to zero. 112 E Homotopy spheres and the KARL invariant

E.3 The Hill–Hopkins–Ravenel result and its inﬂuence on the KARL invariant

Theorem E.9 (Browder [18] and Hill–Hopkins–Ravenel [47]). The “boundary” homomorphism b : P4k+2 = Z/2 → Θ4k+1 is trivial if 4k + 2 = 2, 6, 14, 30, 62, and possibly 126. (These are the exceptional integers +1.) In all other cases it is injective and hence bP4k+2 = Z/2.

Theorem E.10. If (4k + 1) is not exceptional, the Arf invariant of the quadratic form deﬁned via the Seifert form (i.e., the KARL invariant of the knot) detects which homotopy sphere of dimension 4k + 1 is embedded.

Proof of Theorem E.9. Let us first suppose that the knot K4k+1 is simple. Let F4k+2 be a 2k-connected Seifert hypersurface. Hence the quadratic form H2k+1(F; Z/2) → Z/2 “à la Kervaire–Milnor” is defined (see [72, pp. 202–207]). By [68, lemma on normal bundles, p. 512], this last quadratic form coincides with the quadratic form that gives rise to the KARL invariant. Hence KARL(K) detects which element of P4k+2 is represented by F4k+2 and hence which element of bP4k+2 is represented by K4k+1. Let us now withdraw the assumption that K is simple. We know that its KARL invariant is defined. By [80] there exists a simple knot K∗ that is knot-cobordant to K. By [79] the knots K and K∗ have the same KARL invariant. Since K and K∗ are diffeomorphic, KARL(K∗) = KARL(K) detects which element of bP4k+2 is represented by K.

Recap. In terms of bP4k+2 the interpretation of the KARL invariant splits into two cases.

(i) If bP4k+2 = Z/2 there are two homotopy spheres of dimension (4k +1) that can be embedded in S4k+3 to represent knots. The KARL invariant detects which homotopy sphere it is. Thus the differential structure on the homotopy sphere is detected by the Alexander module (in dimension (2k + 1)) of the knot. In other words, different differential structures give rise to different knot modules in dimension (2k + 1).

Caution. The KARL invariant detects the Arf invariant of the handlebody (i.e., of the Seifert hypersurface). If you know beforehand that the Kervaire invariant is nontrivial, then it detects which homotopy sphere is represented by the boundary of the handlebody. E.3 The Hill–Hopkins–Ravenel result and the KARL invariant 113

(ii) If bP4k+2 = 0, all (4k + 1)-knots are represented by an embedding of the standard sphere. From a knot theory point of view, one could argue that this situation is more interesting, since the KARL invariant is a genuine invariant of the embedding. The case of classical knots is typical. An (easy) consequence of our discussion of the KARL invariant produces a proof of the following result, when n = 4k + 1. If n = 4k + 3, it can be obtained by using the signature and Blanchﬁeld duality. See Dieter Erle [33].

Theorem E.11. Let n ≥ 5. Let Σn and bΣn be homotopy spheres embedded in Sn+2. Suppose that the two homotopy spheres are not diﬀeomorphic. Then the two knots are not topologically isotopic.

Figures

In this chapter we propose a few figures to help to visualize some concepts. We find it useful to gather the figures together in the same place, since the related concepts are spread out in the text. In AppendixA, we presented the general conventions for orientation and sign that we have followed throughout the book. These are valid in any dimension. In Figures F.1, F.2, and F.3 we show what we consider to be the standard orientations of the 2-plane and 3-space. This enables us to ascribe a sign to some objects represented in the pictures. We are aware that these conventions in small dimensions are subject to controversy.

1 2 Figure F.1. S is oriented as the boundary of B . In the ﬁgure the frame (e1, e2) represents our standard orientation of the plane.

e2 e e3 1

Figure F.2. Space orientation braid representation of a Hopf link. If oriented as a closed braid, the linking coefficient between the two components is equal to +1, since the frame (e1, e2, e3) corresponds to the three fingers of the right hand: (thumb, forefinger, middle finger). The frame is obtained as follows. The vectors (e1, e2) are tangent to the 2-disc spanned by the component that is on the right. The vector e3 is tangent to the component that is on the left. 116 F Figures

+ +

2 _

1 Figure F.3. Linking number in S . We have LS1 (∂γ1, ∂γ2) = −1 since IB2 (γ1, γ2) = −1.

meridian circle

Figure F.4. Torus.

Figure F.5. 2-sphere with corners.

Key to Figures F.4 and F.5: Surgery on the torus along a meridian circle. The result of the surgery is a 2-sphere with corners. F Figures 117

Figure F.6. Seifert surface for the Hopf link represented in Figure F.2.

Figure F.7. Seifert surface for a trefoil knot presented as a 1-handlebody in 3-space.

e e2 1 a2 a1

Figure F.8. How to obtain the Seifert form of the trefoil of Figure F.7. 118 F Figures

Figure F.9. The tube and the crooked ball as in Milnor’s book, p. 98.

Figure F.10. A plumbing in dimension 2. F Figures 119

Figure F.11. One of the two possible parallelizable 1-handlebodies with two handles.

Figure F.12. The other of the two possible parallelizable 1-handlebodies with two handles.

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Bn+1 standard ball of dimension (n + 1) 9

B˚n+1 standard open ball of dimension (n + 1) 9 bPn+1 group of homotopy n-spheres that bound parallelizable manifolds 18

C2q cobordism group of 2q-dimensional knots 69

C2q−1 cobordism group of (2q − 1)-dimensional knots 71

C cobordism group of -forms 72

Γn group of twisted n-spheres 19

Λn set of oriented homotopy n-spheres 15

Pn+1 Kervaire–Milnor framed cobordism group 22

Sn standard n-sphere9

Θn h-cobordism group of oriented homotopy n-spheres 16

Index of Quoted Theorems

Adams theorem on the stable J-homomorphism 13

Bott periodicity theorem 13

Brieskorn theorem on Pham–Brieskorn singularities5

Haeﬂiger theorem on diﬀerentiable knots of S4k−1 in S6k 29

Haeﬂiger theorem on embeddings in the metastable range 29

Haeﬂiger theorem on m-dimensional spherical links in S2m+1 103

Hill–Hopkins–Ravenel theorem on the Kervaire invariant 22, 112

Hopf theorem on H2(Γ) 36

Kervaire’s conditions on a knot group 35

Kervaire theorem on the realizabilty of his conditions on a knot group 38

Kervaire theorem on knot modules 48

Kervaire–Milnor main theorem on Θn 17

Kervaire–Milnor theorem on bPn+1 18

Kervaire–Milnor’s computation of Pn+1 23

Levine unknotting theorem 43

Levine theorem on S-equivalence for simple knots 67

Levine theorem on the algebraic characterization of the 73 knot-cobordism group

Milnor ﬁbration theorem 76

Mumford theorem on singularities of complex algebraic surfaces 81

Pontrjagin isomorphism theorem 12 132 Index of Quoted Theorems

Realization theorem of Seifert matrices 62

Recognition theorem of q-handlebodies 100

Silver–Whitten–Williams theorem on killers in a knot group 36

Whitehead theorem on the homotopy type of C.W. complexes 43

Whitney theorem on embeddings in the stable range 28 Index of Terminology

algebraic link, 79 Kervaire–Arf–Robertello–Levine almost parallelizable manifold, 16 invariant, 111 Arf invariant, 21 Kervaire conjecture, 40 aspherical C.W. complex, 43 Kervaire invariant, 21 attaching link, 61, 99 Kervaire manifold, 25 Kervaire’s Paris paper,1 Bernoulli number Bk, 13 knot,1 binding (of an open book knot manifold, 37 decomposition), 76, 95 knot module, 44 Bott periodicity, 13 link of a germ, 75 classical knot, 33 link, n-link in codimension q,3 closed manifold,7 linking number, 92 cobordant -forms, 72 cobordant knots, 69 m-spherical r-link in S2m+1, 103 codimension of a link,3 mapping cylinder, 85 common part of a surgery,9 mapping torus, 76, 95 core, 61, 99 meridian of a knot, 36 meridian of a link, 93 -form, 72 metabolizer of an -form, 72 exterior of a knot,1, 33 Milnor ﬁber, 75 exterior of a submanifold,7 minimal (Seifert hypersurface), 52 framed manifold, 12 module of type K, 46 framed cobordant manifolds, 12 monodromy, 76 framed cobordism, 23 n-knot in codimension q,4, 27 group of a knot,4 null-cobordant -form, 72 group of a link,4 null-cobordant knot, 69 h-cobordant manifolds, 29 open book decomposition, 76, 95 handle of index q, 99 ordinary double point, 83 handle presentation, 61, 99 page (of an open book decomposition), handlebody, q-handlebody, 24, 61, 99 76 homotopy sphere, 15 parallelizable manifold, 10 intersection number, 92 Pham–Brieskorn singularity, 83 plumbing, 104 join (of two compact spaces), 85 Pontrjagin construction, 12 134 Index of Terminology rank of a vector bundle,7

S-equivalent (Seifert forms), 55 scars of a surgery,9 Seifert form, 53 Seifert hypersurface of a link,4, 93 Seifert hypersurface of a knot, 34, 50 (2q − 1)-simple ﬁbered link, 77 (2q − 1)-simple link, 61 simple n-knot, 52 skeleton (of a q-handlebody), 61 slice knot, 69 stable homotopy group of spheres, 12 stable J-homomorphism, 13 stably parallelizable manifold, 10 stably trivial vector bundle, 10 standard ball,9 standard n-knot, 33 standard sphere,9 surgery,9 topological type (of a germ), 75 topologically trivial germ, 82 torsion linking forms, 57 torsion Seifert forms, 58 trivial vector bundle, 10 weight (of a group), 35 Series of Lectures in Mathematics

Françoise Michel Françoise Michel and Claude Weber Claude Weber Françoise Michel Higher-Dimensional Knots Claude Weber According to Michel Kervaire

Michel Kervaire wrote six papers on higher-dimensional knots which can be considered fundamental to the development of the theory. They are not

only of historical interest but naturally introduce to some of the essential Higher-Dimensional Knots According to Michel Kervaire Higher-Dimensional techniques in this fascinating area.

ISBN 978-3-03719-180-4

www.ems-ph.org

Michel/Weber | Rotis Sans | Pantone 287, Pantone 116 | 170 x 240 mm | RB: 7.2 mm