On the Browder-Levine-Novikov Embedding Theorems 3
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On the Stable Classification of Certain 4-Manifolds
BULL. AUSTRAL. MATH. SOC. 57N65, 57R67, 57Q10, 57Q20 VOL. 52 (1995) [385-398] ON THE STABLE CLASSIFICATION OF CERTAIN 4-MANIFOLDS ALBERTO CAVICCHIOLI, FRIEDRICH HEGENBARTH AND DUSAN REPOVS We study the s-cobordism type of closed orientable (smooth or PL) 4—manifolds with free or surface fundamental groups. We prove stable classification theorems for these classes of manifolds by using surgery theory. 1. INTRODUCTION In this paper we shall study closed connected (smooth or PL) 4—manifolds with special fundamental groups as free products or surface groups. For convenience, all manifolds considered will be assumed to be orientable although our results work also in the general case, provided the first Stiefel-Whitney classes coincide. The starting point for classifying manifolds is the determination of their homotopy type. For 4- manifolds having finite fundamental groups with periodic homology of period four, this was done in [12] (see also [1] and [2]). The case of a cyclic fundamental group of prime order was first treated in [23]. The homotopy type of 4-manifolds with free or surface fundamental groups was completely classified in [6] and [7] respectively. In particular, closed 4-manifolds M with a free fundamental group IIi(M) = *PZ (free product of p factors Z ) are classified, up to homotopy, by the isomorphism class of their intersection pairings AM : B2{M\K) x H2(M;A) —> A over the integral group ring A = Z[IIi(M)]. For Ui(M) = Z, we observe that the arguments developed in [11] classify these 4-manifolds, up to TOP homeomorphism, in terms of their intersection forms over Z. -
Finite Group Actions on Kervaire Manifolds 3
FINITE GROUP ACTIONS ON KERVAIRE MANIFOLDS DIARMUID CROWLEY AND IAN HAMBLETON M4k+2 Abstract. Let K be the Kervaire manifold: a closed, piecewise linear (PL) mani- fold with Kervaire invariant 1 and the same homology as the product S2k+1 × S2k+1 of M4k+2 spheres. We show that a finite group of odd order acts freely on K if and only if 2k+1 2k+1 it acts freely on S × S . If MK is smoothable, then each smooth structure on M j M4k+2 K admits a free smooth involution. If k 6= 2 − 1, then K does not admit any M30 M62 free TOP involutions. Free “exotic” (PL) involutions are constructed on K , K , and M126 M30 Z Z K . Each smooth structure on K admits a free /2 × /2 action. 1. Introduction One of the main themes in geometric topology is the study of smooth manifolds and their piece-wise linear (PL) triangulations. Shortly after Milnor’s discovery [54] of exotic smooth 7-spheres, Kervaire [39] constructed the first example (in dimension 10) of a PL- manifold with no differentiable structure, and a new exotic smooth 9-sphere Σ9. The construction of Kervaire’s 10-dimensional manifold was generalized to all dimen- sions of the form m ≡ 2 (mod 4), via “plumbing” (see [36, §8]). Let P 4k+2 denote the smooth, parallelizable manifold of dimension 4k+2, k ≥ 0, constructed by plumbing two copies of the the unit tangent disc bundle of S2k+1. The boundary Σ4k+1 = ∂P 4k+2 is a smooth homotopy sphere, now usually called the Kervaire sphere. -
Mapping Surgery to Analysis III: Exact Sequences
K-Theory (2004) 33:325–346 © Springer 2005 DOI 10.1007/s10977-005-1554-7 Mapping Surgery to Analysis III: Exact Sequences NIGEL HIGSON and JOHN ROE Department of Mathematics, Penn State University, University Park, Pennsylvania 16802. e-mail: [email protected]; [email protected] (Received: February 2004) Abstract. Using the constructions of the preceding two papers, we construct a natural transformation (after inverting 2) from the Browder–Novikov–Sullivan–Wall surgery exact sequence of a compact manifold to a certain exact sequence of C∗-algebra K-theory groups. Mathematics Subject Classifications (1991): 19J25, 19K99. Key words: C∗-algebras, L-theory, Poincare´ duality, signature operator. This is the final paper in a series of three whose objective is to construct a natural transformation from the surgery exact sequence of Browder, Novikov, Sullivan and Wall [17,21] to a long exact sequence of K-theory groups associated to a certain C∗-algebra extension; we finally achieve this objective in Theorem 5.4. In the first paper [5], we have shown how to associate a homotopy invariant C∗-algebraic signature to suitable chain complexes of Hilbert modules satisfying Poincare´ duality. In the second paper, we have shown that such Hilbert–Poincare´ complexes arise natu- rally from geometric examples of manifolds and Poincare´ complexes. The C∗-algebras that are involved in these calculations are analytic reflections of the equivariant and/or controlled structure of the underlying topology. In paper II [6] we have also clarified the relationship between the analytic signature, defined by the procedure of paper I for suitable Poincare´ com- plexes, and the analytic index of the signature operator, defined only for manifolds. -
Surgery on Compact Manifolds, Second Edition, 1999 68 David A
http://dx.doi.org/10.1090/surv/069 Selected Titles in This Series 69 C. T. C. Wall (A. A. Ranicki, Editor), Surgery on compact manifolds, Second Edition, 1999 68 David A. Cox and Sheldon Katz, Mirror symmetry and algebraic geometry, 1999 67 A. Borel and N. Wallach, Continuous cohomology, discrete subgroups, and representations of reductive groups, Second Edition, 1999 66 Yu. Ilyashenko and Weigu Li, Nonlocal bifurcations, 1999 65 Carl Faith, Rings and things and a fine array of twentieth century associative algebra, 1999 64 Rene A. Carmona and Boris Rozovskii, Editors, Stochastic partial differential equations: Six perspectives, 1999 63 Mark Hovey, Model categories, 1999 62 Vladimir I. Bogachev, Gaussian measures, 1998 61 W. Norrie Everitt and Lawrence Markus, Boundary value problems and symplectic algebra for ordinary differential and quasi-differential operators, 1999 60 Iain Raeburn and Dana P. Williams, Morita equivalence and continuous-trace C*-algebras, 1998 59 Paul Howard and Jean E. Rubin, Consequences of the axiom of choice, 1998 58 Pavel I. Etingof, Igor B. Frenkel, and Alexander A. Kirillov, Jr., Lectures on representation theory and Knizhnik-Zamolodchikov equations, 1998 57 Marc Levine, Mixed motives, 1998 56 Leonid I. Korogodski and Yan S. Soibelman, Algebras of functions on quantum groups: Part I, 1998 55 J. Scott Carter and Masahico Saito, Knotted surfaces and their diagrams, 1998 54 Casper Goffman, Togo Nishiura, and Daniel Waterman, Homeomorphisms in analysis, 1997 53 Andreas Kriegl and Peter W. Michor, The convenient setting of global analysis, 1997 52 V. A. Kozlov, V. G. Maz'ya> and J. Rossmann, Elliptic boundary value problems in domains with point singularities, 1997 51 Jan Maly and William P. -
Knots, 3-Manifolds and the Lickorish Wallace Theorem
KNOTS, 3-MANIFOLDS AND THE LICKORISH WALLACE THEOREM A REPORT submitted in partial fulfillment of the requirements for the award of the dual degree of Bachelor of Science-Master of Science in MATHEMATICS by M V AJAY KUMAR NAIR (13076) DEPARTMENT OF MATHEMATICS INDIAN INSTITUTE OF SCIENCE EDUCATION AND RESEARCH BHOPAL BHOPAL - 462066 April 2018 i CERTIFICATE This is to certify that M V Ajay Kumar Nair, BS-MS (Dual Degree) student in Department of Mathematics has completed bonafide work on the dissertation entitled `Knots, 3-manifolds and the Lickorish Wallace theorem' under my supervision and guidance. April 2018 Dr. Kashyap Rajeevsarathy IISER Bhopal Committee Member Signature Date ii ACADEMIC INTEGRITY AND COPYRIGHT DISCLAIMER I hereby declare that this project is my own work and, to the best of my knowledge, it contains no materials previously published or written by another person, or substantial proportions of material which have been ac- cepted for the award of any other degree or diploma at IISER Bhopal or any other educational institution, except where due acknowledgement is made in the document. I certify that all copyrighted material incorporated into this document is in compliance with the Indian Copyright (Amendment) Act (2012) and that I have received written permission from the copyright owners for my use of their work, which is beyond the scope of the law. I agree to indemnify and saveguard IISER Bhopal from any claims that may be arise from any copyright violation. April 2018 M V Ajay Kumar Nair IISER Bhopal iii ACKNOWLEDGEMENT I would like to extend my heartfelt gratitude to my thesis supervisor Dr Kashyap Rajeevsarathy for the persistent inspiration and guidance. -
Milnor, John W. Groups of Homotopy Spheres
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 52, Number 4, October 2015, Pages 699–710 http://dx.doi.org/10.1090/bull/1506 Article electronically published on June 12, 2015 SELECTED MATHEMATICAL REVIEWS related to the paper in the previous section by JOHN MILNOR MR0148075 (26 #5584) 57.10 Kervaire, Michel A.; Milnor, John W. Groups of homotopy spheres. I. Annals of Mathematics. Second Series 77 (1963), 504–537. The authors aim to study the set of h-cobordism classes of smooth homotopy n-spheres; they call this set Θn. They remark that for n =3 , 4thesetΘn can also be described as the set of diffeomorphism classes of differentiable structures on Sn; but this observation rests on the “higher-dimensional Poincar´e conjecture” plus work of Smale [Amer. J. Math. 84 (1962), 387–399], and it does not really form part of the logical structure of the paper. The authors show (Theorem 1.1) that Θn is an abelian group under the connected sum operation. (In § 2, the authors give a careful treatment of the connected sum and of the lemmas necessary to prove Theorem 1.1.) The main task of the present paper, Part I, is to set up methods for use in Part II, and to prove that for n = 3 the group Θn is finite (Theorem 1.2). (For n =3the authors’ methods break down; but the Poincar´e conjecture for n =3wouldimply that Θ3 = 0.) We are promised more detailed information about the groups Θn in Part II. The authors’ method depends on introducing a subgroup bPn+1 ⊂ Θn;asmooth homotopy n-sphere qualifies for bPn+1 if it is the boundary of a parallelizable man- ifold. -
Arxiv:1506.05408V1 [Math.AT]
NOVIKOV’S CONJECTURE JONATHAN ROSENBERG Abstract. We describe Novikov’s “higher signature conjecture,” which dates back to the late 1960’s, as well as many alternative formulations and related problems. The Novikov Conjecture is perhaps the most important unsolved problem in high-dimensional manifold topology, but more importantly, vari- ants and analogues permeate many other areas of mathematics, from geometry to operator algebras to representation theory. 1. Origins of the Original Conjecture The Novikov Conjecture is perhaps the most important unsolved problem in the topology of high-dimensional manifolds. It was first stated by Sergei Novikov, in various forms, in his lectures at the International Congresses of Mathematicians in Moscow in 1966 and in Nice in 1970, and in a few other papers [84, 87, 86, 85]. For an annotated version of the original formulation, in both Russian and English, we refer the reader to [37]. Here we will try instead to put the problem in context and explain why it might be of interest to the average mathematician. For a nice book- length exposition of this subject, we recommend [65]. Many treatments of various aspects of the problem can also be found in the many papers in the collections [38, 39]. For the typical mathematician, the most important topological spaces are smooth manifolds, which were introduced by Riemann in the 1850’s. However, it took about 100 years for the tools for classifying manifolds (except in dimension 1, which is trivial, and dimension 2, which is relatively easy) to be developed. The problem is that manifolds have no local invariants (except for the dimension); all manifolds of the same dimension look the same locally. -
Geometric Periodicity and the Invariants of Manifolds
GEOMETRIC PERIODICITY AND THE INVARIANTS OF MANIFOLDS Dennis Sullivan There is a surprisingly rich structure in the theory of manifolds within a given homotovy type, We begin with a geometric procedure for constructing an isomorphism between two manifolds - either smooth, combinatorial, or topological - in the homotopy class of a given hcmlotopy equivalence between them. This method which combines transversality and surgery inside these manifolds leads to three natural obstruction theories for the three situations. The obstructions up to codimension two are stable in a certain sense and much of our discuss}nn concerns them. The unstable eodimenslon two obstruction has not been analyzed. The interesting structure in these theories ariseson the one hand from the natural invariants which can be associated to a homotopy equiyalence M --f-f> L . These enable us to obtain a-priorl information about the obstructions encountered in constructing an isomorphism between ~ and L . By transversality the "varieties in L" can be pulled back to "varieties in M" . This correspondence is a rather deep geometric invariant of f . By a charac- teristic invariant of f we mean any invariant defined using the induced diagram f M >L l l V + L is a variety in L . f_iv fV -> V For example when V is a manifold fV has a well defined surgery obstruction like I/8 (signature f-IV - signature V) ~ Z or Z/n , dim V = 4i ~(f, V) ]Arf invariant (kernel fJ , Z/2)~ Z/2, dim V = 4i + ~ . Consider the homotopy equivalence M f > L where M and L are simply connected and have dimension at least five. -
Manifold Aspects of the Novikov Conjecture
Manifold aspects of the Novikov Conjecture James F. Davis§ 4 Let LM H §(M; Q) be the Hirzebruch L-class of an oriented manifold M. Let Bº2 (or K(º, 1)) denote any aspherical space with fundamental group º. (A space is aspherical if it has a contractible universal cover.) In 1970 Novikov made the following conjecture. Novikov Conjecture. Let h : M 0 M be an orientation-preserving ho- motopy equivalence between closed,! oriented manifolds.1 For any discrete group º and any map f : M Bº, ! f h (LM [M 0]) = f (LM [M]) H (Bº; Q) § ± § 0 \ § \ 2 § Many surveys have been written on the Novikov Conjecture. The goal here is to give an old-fashioned point of view, and emphasize connections with characteristic classes and the topology of manifolds. For more on the topology of manifolds and the Novikov Conjecture see [58], [47], [17]. This article ignores completely connections with C§-algebras (see the articles of Mishchenko, Kasparov, and Rosenberg in [15]), applications of the Novikov conjecture (see [58],[9]), and most sadly, the beautiful work and mathemat- ical ideas uncovered in proving the Novikov Conjecture in special cases (see [14]). The level of exposition in this survey starts at the level of a reader of Milnor-StasheÆ’s book Characteristic Classes, but by the end demands more topological prerequisites. Here is a table of contents: § Partially supported by the NSF. This survey is based on lectures given in Mainz, Germany in the Fall of 1993. The author wishes to thank the seminar participants as well as Paul Kirk, Chuck McGibbon, and Shmuel Weinberger for clarifying conversations. -
Kreck-Stolz Invariants for Quaternionic Line Bundles
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 365, Number 6, June 2013, Pages 3193–3225 S 0002-9947(2012)05732-1 Article electronically published on November 20, 2012 KRECK-STOLZ INVARIANTS FOR QUATERNIONIC LINE BUNDLES DIARMUID CROWLEY AND SEBASTIAN GOETTE Abstract. We generalise the Kreck-Stolz invariants s2 and s3 by defining a new invariant, the t-invariant, for quaternionic line bundles E over closed spin- 3 4 manifolds M of dimension 4k−1withH (M; Q) = 0 such that c2(E) ∈ H (M) is torsion. The t-invariant classifies closed smooth oriented 2-connected ratio- nal homology 7-spheres up to almost-diffeomorphism, that is, diffeomorphism up to a connected sum with an exotic sphere. It also detects exotic homeo- morphisms between such manifolds. The t-invariant also gives information about quaternionic line bundles over a fixed manifold, and we use it to give a new proof of a theorem of Feder and Gitler about the values of the second Chern classes of quaternionic line bundles over HP k.Thet-invariant for S4k−1 is closely related to the Adams e-invariant on the (4k − 5)-stem. Introduction In [25], Kreck and Stolz introduced invariants s1, s2, s3 ∈ Q/Z of certain closed smooth oriented simply connected 7-manifolds M that completely characterise M up to diffeomorphism. The s-invariants are defect invariants based on index theo- rems for Dirac operators on 8-manifolds: the invariant s1, which equals the Eells- Kuiper invariant μ of [17] if M is spin, is the defect of the untwisted Dirac operator, whereas s2 and s3 are the defects of the Dirac operator twisted by certain complex line bundles. -
Math 2701 - Topology 2 Possible Final Project Topics List
Math 2701 - Topology 2 Possible Final Project Topics List The classification of one-manifolds. There do not seem to be many proofs of this classical result in print, though there are more on the internet. Guillemin{Pollack's Differential Topology gives one in the smooth case, and David Gale's American Mathematical Monthly article \The Classification of 1-Manifolds: A Take-Home Exam" gives a topological proof (in exercises; google it). The classification of surfaces. A cornerstone result in many different areas of topology. There's a classical proof using triangulations which is presented, for instance, in Massey's A Basic Course in Algebraic Topology, Chapter 1 x7. \Conway's ZIP Proof" (American Mathematical Monthly article by Francis{Weeks; google it) presents a \lighter, fat-free nouvelle cuisine approach that retains all the classical flavor of elementary topology", due to John H. Conway. The Jordan Curve theorem, the Schoenfliess Theorem, and the Alexander horned 2 sphere. The first asserts that every embedded circle in R divides it into two components, each of which it bounds; the second, that one of these is homeomorphic to a disk. These are discussed in most basic topology books, eg. Armstrong's Basic Topology and Munkres' Topology. The Jordan curve theorem extends to higher dimensions whereas the Schoenfliess theorem does not (in the topological category, at least), and the Alexander horned sphere is a counterexample: a two-sphere 3 embedded in R that does not bound a ball. See Hatcher's Algebraic Topology, Example 2B.2. First homology vs. the abelianization of π1. Section 1.A of Hatcher's Algebraic Topology gives a proof of the fundamental computational result that for a path connected space X, H1(X) ab is naturally isomorphic to the abelianization π1(X) of π1(X). -
A Composition Formula for Manifold Structures
1 A COMPOSITION FORMULA FOR MANIFOLD STRUCTURES Andrew Ranicki (Edinburgh) Paper: http://www.maths.ed.ac.uk/eaar/papers/compo.pdf Slides: http://www.maths.ed.ac.uk/eaar/slides/bonn.pdf Bonn, 4th December, 2007 2 Homotopy equivalences and homeomorphisms I Every homotopy equivalence of 2-dimensional manifolds is homotopic to a homeomorphism. I For n > 3 a homotopy equivalence of n-dimensional manifolds f : N ! M is not in general homotopic to a homeomorphism, e.g. lens spaces for n = 3. I There are surgery obstructions to making the normal maps f j : f −1(L) ! L normal bordant to homotopy equivalences for every submanifold L ⊂ M. For n > 5 f is homotopic to a homeomorphism if and only if there exist such normal bordisms which are compatible with each other. I Novikov (1964) used surgery to construct homotopy p q equivalences f : N ! M = S × S for certain p; q > 2, which are not homotopic to homeomorphisms. 3 The topological structure set STOP (M) TOP I The structure set S (M) of an n-dimensional topological manifold M is the pointed set of equivalence classes of pairs (N; f ) with N an n-dimensional topological manifold and f : N ! M a homotopy equivalence. 0 0 −1 0 0 I (N; f ) ∼ (N ; f ) if (f )f : N ! N is homotopic to a homeomorphism. TOP I Base point (M; 1) 2 S (M). TOP n I Poincar´econjecture S (S ) = {∗}. TOP I Borel conjecture If M is aspherical then S (M) = {∗}. TOP I For n > 5 surgery theory expresses S (M) in terms of topological K-theory of the homotopy type of M and TOP algebraic L-theory of Z[π1(M)].