DOCSLIB.ORG

The Algebraic Surgery Exact Sequence

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

The algebraic surgery exact sequence ANDREW RANICKI Edinburgh http://www.maths.ed.ac.uk/ aar e algebraic surgery exact sequence • The de ned for any space is X A ; [ ] Hn X L Ln Z π1 X ···→ • −→ ; Sn X Hn 1 X L ... → → − • → the -theory assembly map. The with A L functor is homotopy invariant. X S X 7→ ∗ 2-stage obstructions of the Browder- • The Novikov-Sullivan-Wall surgery theory for the man- existence and uniqueness of top ological ifold structures in a homotopy typ e are re- placed by single obstructions in the relative groups of the assembly map . S X A ∗ 1 Lo cal and global mo dules The assembly map : ; [ ] A H X L L Z π1 X • ∗ • → ∗ is induced by a forgetful functor : -mo dules [ ]-mo dules A Z,X Z π1 X { }→{ } where the domain dep ends on the lo cal and the target dep ends only of top ology X on the fundamental group of , which π1 X X is global. ! with terms of sheaf theory = In A q p • ! the universal covering projection p : X X → and pt. q : X f →{ } f geometric mo del for the -theory as- The L • is the forgetful functor sembly A Poincar e complexes {geometric } manifolds top ological . →{ } fact, in dimensions 5 this functor In n ≥ has the same bre as A. 2 Lo cal and global quadratic Poincar e complexes Global The -group [ ] is the L Ln Z π1 X • rdism group of -dimensional quadratic cob o n Poincar e complexes over [ ]. C, ψ Z π1 X The generalized homology group • Lo cal ; is the cob ordism group of - Hn X L n • dimensional quadratic Poincar e complexes over . As in sheaf theory C, ψ Z,X C stalks, which are -mo dule chain com- has Z . plexes C x x X ∈ is the cob ordism group Lo cal-Global Sn X • of quadratic Poincar e n 1-dimensional − over such that the complexes C, ψ Z,X [ ]-mo dule chain complex assembly Z π1 X is acyclic. AC 3 The total surgery obstruction total surgery obstruction of an • The geometric Poincar e complex n-dimensional the cob ordism class X is = sX C, ψ Sn X ∈ of a [ ]-acyclic 1-dimensional Z π1 X n − quadratic Poincar e complex over . C, ψ Z,X measure the fail- stalks The C x x X ∈ ure of have lo cal Poincar e duality X to n r Hr C x H x Hr X, X x ···→ → − { } → \{ } n r +1 Hr 1 C x H − x ... → − → { } → an -dimensional homology manifold if X is n = 0. In particular, this only if and H C x ∗ the case if is a top ological manifold. is X otal Surgery Obstruction Theorem • T = 0 if and for 5 only if sX Sn X n ∈ ≥ homotopy equivalent to an -dimensional X is n top ological manifold. 4 The pro of of the Total Surgery Obstruction Theorem The pro of is a translation into algebra of the two-stage Browder-Novikov-Sullivan-Wall ob- struction for the existence of a top ological man- ifold in the homotopy typ e of a Poincar e com- : plex X The image ; of is t X Hn 1 X L s X • ∈ − • that = 0 if and only if the Spi- such t X normal bration : has a vak ν X BG X → top ological reduction . ν : X BTOP X → e = 0 then is the im- If t X s X Sn X • ∈ of the surgery obstruction age σ f, b ∗ ∈ [ ] of the normal map : Ln Z π1 X f M → determined by a choice of X, b : ν ν M → X : . lift νX X BTOP →e e = 0 if and only if there exists a reduc- sX • : for which = 0. tion νX X BTOP σ f, b → ∗ e 5 The structure invariant structure invariant of a homotopy equiv- • The : of -dimensional top o- alence h N M n → logical manifolds is the cob ordism class = sh C, ψ M Sn+1 ∈ a globally acyclic -dimensional quadratic of n Poincar e complex The stalks C, ψ . C x measure the failure of to have x M h ∈ acyclic p oint inverses, with 1 = H C x H h− x x ∗ ∗ →{ } 1 = H h x x M . +1 − ∗ ∈ f acyclic p oint inverses if and only if h has • = 0. In particular, this is the H C x ∗ is a homeomorphism. if case h Invariant Theorem • Structure = 0 if and for 5 only sh M n Sn+1 ∈ ≥ is homotopic to a homeomorphism. if h 6 The pro of of the Structure Invariant Theorem I The pro of is a translation into algebra of the two-stage Browder-Novikov-Sullivan-Wall ob- struction for the uniqueness of top ological man- ifold structures in a homotopy typ e : image ; of is such the t h Hn M L s h • ∈ • = 0 if and only if the normal in- that t h variant can be trvialized 1 : h− ∗νN νM M L0 G/T OP − ' {∗} → ' and only if 1 : extends if h M N M M ∪ ∪ → ∪ to a normal bordism : ; [0 1]; 0 1 f, b W M,N M , , → × { } { } = 0 then is the if t h s h M Sn+1 • ∈ image of the surgery obstruction [ ] σ f, b L π1 M . n+1 Z ∗ ∈ 7 The pro of of the Structure Invariant Theorem I I = 0 if and only if there exists a normal sh • rdism which is a simple homotopy bo f, b equivalence. to work with simple -groups here, Have L • to take advantage of the rdism the- s-cob o orem. pro of. The mapping cylinder • Alternative : of h N M → [0 1] P = M N , ∪h × + 1-dimensional geometric an de nes n Poincar e pair with manifold b ound- P, M N ∪ ry, such that is homotopy equivalent to a P The structure invariant is the rel to- M . ∂ tal surgery obstruction = = sh s P P M . +1 ∂ Sn+1 Sn ∈ 8 The simply-connected case For = 1 the algebraic surgery ex- π1 X • { } act sequence breaks up 0 ; 0 1 Sn X Hn 1 X L Ln Z → → − • → − → total surgery obstruction The s X Sn X • ∈ injectively to the reducibility ob- maps TOP struction ; of the Spi- t X Hn 1 X L ∈ − • normal bration . Thus for 5 a vak ν n X ≥ simply-connected geometric n-dimensional is homotopy equiva- oincar e complex P X lent to an top ological man- n-dimensional : admits a if and only if ifold ν X BG X → : . TOP reduction ν X BTOP X → e structure invariant The s h M Sn+1 • ∈ maps injectively to the normal invariant ; = [ ]. Thus for th Hn M L M,G/TOP ∈ • is homotopic to a homeomorphism n 5 h ≥ if and only if : . th M G/T OP ' {∗} → 9 The geometric surgery exact sequence TOP structure set of a top olog- The S M • manifold is the set of equivalence ical M of homotopy equivalences : classes h N → top ological manifolds , with M from N there exist a homeomorphism : h h if g ∼ 0 a homotopy : . N N and hg h N M 0 → ' 0 0 → rem Quinn, R. The geometric surgery • Theo = dim 5 sequence for exact n M ≥ TOP [ ] L π1 M M n+1 Z S ···→ → [ ] [ ] M,G/TOP Ln Z π1 M → → is isomorphic to the relevant portion of the algebraic surgery exact sequence [ ] L π1 M M +1 n+1 Z Sn ···→ → A ; [ ] Hn M L Ln Z π1 M → • −→ TOP k k 1 . = with M D ,M S M +1 S Sn+k ∂ × × − TOP n n = = 0. Example S S S Sn+1 10 The image of the assembly map rem For any nitely presented group • Theo image of the assembly map π the 1; [ ] A : Hn K π, L Ln Z π • → the subgroup of [ ] consisting of is Ln Z π surgery obstructions of normal the σ f, b ∗ maps : of closed -dimensional f, b N M n → manifolds with = . π1 M π are many calculations of the image • There for nite , notably the Oozing Con- of A π jecture proved by Hambleton-Milgram-Taylor- Williams. 11 Statement of the Novikov conjecture -genus of an -dimensional manifold The n • L a collection of cohomology classes M is 4 ; which are determined M H M Q L ∈ ∗ y the Pontrjagin classes of : b ν M M → In general, is not a homo- BTOP . M L topy invariant. Hirzebruch signature theorem for a 4 - The k • manifold dimensional M 2k = [ ] signature H M , M , M Z ∪ hL i∈ ws that part of the -genus is homotopy sho L invariant. Novikov conjecture for a discrete group • The that the higher signatures for any man- π is ifold with = M π1 M π = [ ] 1; σx M x M , M Q x H∗ K π, Q h ∪L i∈ ∈ are homotopy invariant. 12 Algebraic formulation of the Novikov conjecture rem The Novikov conjecture holds for • Theo group if and only if the rational assem- a π bly maps : 1; = 1; A Hn K π, L Q Hn 4 K π, Q • ⊗ − ∗ [ ] Ln Z π1 M Q → ⊗ are injective. rivially true for nite . T π • eri ed for many in nite groups using al- V π • gebra, geometric group theory, ras, C∗ -algeb etc. See Pro ceedings of 1993 Ob erwolfach conference LMS Lecture Notes 226,227 for state of the art in 1995, not substan- tially out of date. 13 Statement of the Borel conjecture -dimensional Poincar e duality group An n π • is a discrete group such that the classifying space 1 is an -dimensional Poincar e K π, n complex.
Recommended publications
  • Arxiv:1006.1489V2 [Math.GT] 8 Aug 2010 Ril.Ias Rfie Rmraigtesre Rils[14 Articles Survey the Reading from Profited Also I Article

    Arxiv:1006.1489V2 [Math.GT] 8 Aug 2010 Ril.Ias Rfie Rmraigtesre Rils[14 Articles Survey the Reading from Profited Also I Article

    Pure and Applied Mathematics Quarterly Volume 8, Number 1 (Special Issue: In honor of F. Thomas Farrell and Lowell E. Jones, Part 1 of 2 ) 1—14, 2012 The Work of Tom Farrell and Lowell Jones in Topology and Geometry James F. Davis∗ Tom Farrell and Lowell Jones caused a paradigm shift in high-dimensional topology, away from the view that high-dimensional topology was, at its core, an algebraic subject, to the current view that geometry, dynamics, and analysis, as well as algebra, are key for classifying manifolds whose fundamental group is infinite. Their collaboration produced about fifty papers over a twenty-five year period. In this tribute for the special issue of Pure and Applied Mathematics Quarterly in their honor, I will survey some of the impact of their joint work and mention briefly their individual contributions – they have written about one hundred non-joint papers. 1 Setting the stage arXiv:1006.1489v2 [math.GT] 8 Aug 2010 In order to indicate the Farrell–Jones shift, it is necessary to describe the situation before the onset of their collaboration. This is intimidating – during the period of twenty-five years starting in the early fifties, manifold theory was perhaps the most active and dynamic area of mathematics. Any narrative will have omissions and be non-linear. Manifold theory deals with the classification of ∗I thank Shmuel Weinberger and Tom Farrell for their helpful comments on a draft of this article. I also profited from reading the survey articles [14] and [4]. 2 James F. Davis manifolds. There is an existence question – when is there a closed manifold within a particular homotopy type, and a uniqueness question, what is the classification of manifolds within a homotopy type? The fifties were the foundational decade of manifold theory.
  • Homological Algebra

    Homological Algebra

    Homological Algebra Donu Arapura April 1, 2020 Contents 1 Some module theory3 1.1 Modules................................3 1.6 Projective modules..........................5 1.12 Projective modules versus free modules..............7 1.15 Injective modules...........................8 1.21 Tensor products............................9 2 Homology 13 2.1 Simplicial complexes......................... 13 2.8 Complexes............................... 15 2.15 Homotopy............................... 18 2.23 Mapping cones............................ 19 3 Ext groups 21 3.1 Extensions............................... 21 3.11 Projective resolutions........................ 24 3.16 Higher Ext groups.......................... 26 3.22 Characterization of projectives and injectives........... 28 4 Cohomology of groups 32 4.1 Group cohomology.......................... 32 4.6 Bar resolution............................. 33 4.11 Low degree cohomology....................... 34 4.16 Applications to finite groups..................... 36 4.20 Topological interpretation...................... 38 5 Derived Functors and Tor 39 5.1 Abelian categories.......................... 39 5.13 Derived functors........................... 41 5.23 Tor functors.............................. 44 5.28 Homology of a group......................... 45 1 6 Further techniques 47 6.1 Double complexes........................... 47 6.7 Koszul complexes........................... 49 7 Applications to commutative algebra 52 7.1 Global dimensions.......................... 52 7.9 Global dimension of
  • Derived Functors and Homological Dimension (Pdf)

    Derived Functors and Homological Dimension (Pdf)

    DERIVED FUNCTORS AND HOMOLOGICAL DIMENSION George Torres Math 221 Abstract. This paper overviews the basic notions of abelian categories, exact functors, and chain complexes. It will use these concepts to define derived functors, prove their existence, and demon- strate their relationship to homological dimension. I affirm my awareness of the standards of the Harvard College Honor Code. Date: December 15, 2015. 1 2 DERIVED FUNCTORS AND HOMOLOGICAL DIMENSION 1. Abelian Categories and Homology The concept of an abelian category will be necessary for discussing ideas on homological algebra. Loosely speaking, an abelian cagetory is a type of category that behaves like modules (R-mod) or abelian groups (Ab). We must first define a few types of morphisms that such a category must have. Definition 1.1. A morphism f : X ! Y in a category C is a zero morphism if: • for any A 2 C and any g; h : A ! X, fg = fh • for any B 2 C and any g; h : Y ! B, gf = hf We denote a zero morphism as 0XY (or sometimes just 0 if the context is sufficient). Definition 1.2. A morphism f : X ! Y is a monomorphism if it is left cancellative. That is, for all g; h : Z ! X, we have fg = fh ) g = h. An epimorphism is a morphism if it is right cancellative. The zero morphism is a generalization of the zero map on rings, or the identity homomorphism on groups. Monomorphisms and epimorphisms are generalizations of injective and surjective homomorphisms (though these definitions don't always coincide). It can be shown that a morphism is an isomorphism iff it is epic and monic.
  • On Controlled Assembly Maps

    On Controlled Assembly Maps

    RIMS Kôkyûroku Bessatsu B39 (2013), 197214 On Controlled Assembly Maps By * Masayuki Yamasaki Abstract Theorem 3.9 of [9] says that the L^{-\infty} homology theory and the controlled L^{-\infty} theory of a simplicially stratified control map p:E\rightarrow X are equivalent. Unfortunately the proof given there contains serious errors. In this paper I give a correct statement and a correct proof. §1. Introduction For a covariant functor \mathrm{J}=\{\mathrm{J}_{n}\} from spaces to spectra and a map p : E\rightarrow X, Quinn defined a homology spectrum \mathbb{H}(X;\mathrm{J}(p))[4]. \mathbb{H} \mathrm{J} defines a covariant functor which sends a pair (X, p : E\rightarrow X) to a - spectrum \mathbb{H}(X;\mathrm{J}(p)) . Suppose we are given a covariant functor \mathrm{J} ) which sends a pair (X, p) to a spectrum \mathrm{J}(X;p) . Then we can define a covariant functor, also denoted from to . And then we obtain a \mathrm{J} , spaces spectra by \mathrm{J}(E)=\mathrm{J}(*;E\rightarrow*) homology spectrum \mathbb{H}(X;\mathrm{J}(p)) for a map p:E\rightarrow X . Quinn showed that, if the original functor \mathrm{J} -) satisfies three axioms (the restriction, continuity, and inverse limit axioms) and p is nice ( i.e . it is a stratified system of fibrations [4]), then there is a homotopy equivalence A:\mathbb{H}(X;\mathrm{J}(p))\rightarrow \mathrm{J}(X;p) when X is compact [4, Characterization Theorem, p.421]. If X is non‐compact, we need to consider a locally‐finite homology theory.
  • Lecture 15. De Rham Cohomology

    Lecture 15. De Rham Cohomology

    Lecture 15. de Rham cohomology In this lecture we will show how differential forms can be used to define topo- logical invariants of manifolds. This is closely related to other constructions in algebraic topology such as simplicial homology and cohomology, singular homology and cohomology, and Cechˇ cohomology. 15.1 Cocycles and coboundaries Let us first note some applications of Stokes’ theorem: Let ω be a k-form on a differentiable manifold M.For any oriented k-dimensional compact sub- manifold Σ of M, this gives us a real number by integration: " ω : Σ → ω. Σ (Here we really mean the integral over Σ of the form obtained by pulling back ω under the inclusion map). Now suppose we have two such submanifolds, Σ0 and Σ1, which are (smoothly) homotopic. That is, we have a smooth map F : Σ × [0, 1] → M with F |Σ×{i} an immersion describing Σi for i =0, 1. Then d(F∗ω)isa (k + 1)-form on the (k + 1)-dimensional oriented manifold with boundary Σ × [0, 1], and Stokes’ theorem gives " " " d(F∗ω)= ω − ω. Σ×[0,1] Σ1 Σ1 In particular, if dω =0,then d(F∗ω)=F∗(dω)=0, and we deduce that ω = ω. Σ1 Σ0 This says that k-forms with exterior derivative zero give a well-defined functional on homotopy classes of compact oriented k-dimensional submani- folds of M. We know some examples of k-forms with exterior derivative zero, namely those of the form ω = dη for some (k − 1)-form η. But Stokes’ theorem then gives that Σ ω = Σ dη =0,sointhese cases the functional we defined on homotopy classes of submanifolds is trivial.
  • On the Cohomology of the Finite Special Linear Groups, I

    On the Cohomology of the Finite Special Linear Groups, I

    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF ALGEBRA 54, 216-238 (1978) On the Cohomology of the Finite Special Linear Groups, I GREGORY W. BELL* Fort Lewis College, Durango, Colorado 81301 Communicated by Graham Higman Received April 1, 1977 Let K be a finite field. We are interested in determining the cohomology groups of degree 0, 1, and 2 of various KSL,+,(K) modules. Our work is directed to the goal of determining the second degree cohomology of S&+,(K) acting on the exterior powers of its standard I + l-dimensional module. However, our solution of this problem will involve the study of many cohomology groups of degree 0 and 1 as well. These groups are of independent interest and they are useful in many other cohomological calculations involving SL,+#) and other Chevalley groups [ 11. Various aspects of this problem or similar problems have been considered by many authors, including [l, 5, 7-12, 14-19, 211. There have been many techni- ques used in studing this problem. Some are ad hoc and rely upon particular information concerning the groups in question. Others are more general, but still place restrictions, such as restrictions on the characteristic of the field, the order of the field, or the order of the Galois group of the field. Our approach is general and shows how all of these problems may be considered in a unified. way. In fact, the same method may be applied quite generally to other Chevalley groups acting on certain of their modules [l].
  • Homotopy Invariance of Higher Signatures and 3-Manifold Groups

    Homotopy Invariance of Higher Signatures and 3-Manifold Groups

    HOMOTOPY INVARIANCE OF HIGHER SIGNATURES AND 3-MANIFOLD GROUPS MICHEL MATTHEY, HERVE´ OYONO-OYONO AND WOLFGANG PITSCH Abstract. For closed oriented manifolds, we establish oriented homotopy in- variance of higher signatures that come from the fundamental group of a large class of orientable 3-manifolds, including the “piecewise geometric” ones in the sense of Thurston. In particular, this class, that will be carefully described, is the class of all orientable 3-manifolds if the Thurston Geometrization Conjec- ture is true. In fact, for this type of groups, we show that the Baum-Connes Conjecture With Coefficients holds. The non-oriented case is also discussed. 1. Introduction and statement of the main results We assume all manifolds to be non-empty, pointed (i.e. we fix a base-point), sec- ond countable, Hausdorff and smooth. Given a closed connected oriented manifold m M of dimension m, let [M] denote either orientation classes in Hm(M; Q) and m 4∗ in H (M; Z), and let LM ∈ H (M; Q) be the Hirzebruch L-class of M, which is defined as a suitable rational polynomial in the Pontrjagin classes of M (see [20, pp. 11–12] or [34, Ex. III.11.15]). Denote the usual Kronecker pairing for M, with rational coefficients, by ∗ h .,. i : H (M; Q) × H∗(M; Q) −→ Q . If M is of dimension m = 4k, then the Hirzebruch Signature Theorem (see [20, Thm. 8.2.2] or [34, p. 233]) says that the rational number hLM , [M]i is the signature of the cup product quadratic form 2k 2k 4k H (M; Z) ⊗ H (M; Z) −→ H (M; Z) = Z·[M] =∼ Z , (x, y) 7−→ x ∪ y .
  • Finite Group Actions on Kervaire Manifolds 3

    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

    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.
  • Math 231Br: Algebraic Topology

    Math 231Br: Algebraic Topology

    algebraic topology Lectures delivered by Michael Hopkins Notes by Eva Belmont and Akhil Mathew Spring 2011, Harvard fLast updated August 14, 2012g Contents Lecture 1 January 24, 2010 x1 Introduction 6 x2 Homotopy groups. 6 Lecture 2 1/26 x1 Introduction 8 x2 Relative homotopy groups 9 x3 Relative homotopy groups as absolute homotopy groups 10 Lecture 3 1/28 x1 Fibrations 12 x2 Long exact sequence 13 x3 Replacing maps 14 Lecture 4 1/31 x1 Motivation 15 x2 Exact couples 16 x3 Important examples 17 x4 Spectral sequences 19 1 Lecture 5 February 2, 2010 x1 Serre spectral sequence, special case 21 Lecture 6 2/4 x1 More structure in the spectral sequence 23 x2 The cohomology ring of ΩSn+1 24 x3 Complex projective space 25 Lecture 7 January 7, 2010 x1 Application I: Long exact sequence in H∗ through a range for a fibration 27 x2 Application II: Hurewicz Theorem 28 Lecture 8 2/9 x1 The relative Hurewicz theorem 30 x2 Moore and Eilenberg-Maclane spaces 31 x3 Postnikov towers 33 Lecture 9 February 11, 2010 x1 Eilenberg-Maclane Spaces 34 Lecture 10 2/14 x1 Local systems 39 x2 Homology in local systems 41 Lecture 11 February 16, 2010 x1 Applications of the Serre spectral sequence 45 Lecture 12 2/18 x1 Serre classes 50 Lecture 13 February 23, 2011 Lecture 14 2/25 Lecture 15 February 28, 2011 Lecture 16 3/2/2011 Lecture 17 March 4, 2011 Lecture 18 3/7 x1 Localization 74 x2 The homotopy category 75 x3 Morphisms between model categories 77 Lecture 19 March 11, 2011 x1 Model Category on Simplicial Sets 79 Lecture 20 3/11 x1 The Yoneda embedding 81 x2 82
  • On the Work of Michel Kervaire in Surgery and Knot Theory

    On the Work of Michel Kervaire in Surgery and Knot Theory

    1 ON MICHEL KERVAIRE'S WORK IN SURGERY AND KNOT THEORY Andrew Ranicki (Edinburgh) http://www.maths.ed.ac.uk/eaar/slides/kervaire.pdf Geneva, 11th and 12th February, 2009 2 1927 { 2007 3 Highlights I Major contributions to the topology of manifolds of dimension > 5. I Main theme: connection between stable trivializations of vector bundles and quadratic refinements of symmetric forms. `Division by 2'. I 1956 Curvatura integra of an m-dimensional framed manifold = Kervaire semicharacteristic + Hopf invariant. I 1960 The Kervaire invariant of a (4k + 2)-dimensional framed manifold. I 1960 The 10-dimensional Kervaire manifold without differentiable structure. I 1963 The Kervaire-Milnor classification of exotic spheres in dimensions > 4 : the birth of surgery theory. n n+2 I 1965 The foundation of high dimensional knot theory, for S S ⊂ with n > 2. 4 MATHEMATICAL REVIEWS + 1 Kervaire was the author of 66 papers listed (1954 { 2007) +1 unlisted : Non-parallelizability of the n-sphere for n > 7, Proc. Nat. Acad. Sci. 44, 280{283 (1958) 619 matches for "Kervaire" anywhere, of which 84 in title. 18,600 Google hits for "Kervaire". MR0102809 (21() #1595) Kervaire,, Michel A. An interpretation of G. Whitehead's generalizationg of H. Hopf's invariant. Ann. of Math. (2)()69 1959 345--365. (Reviewer: E. H. Brown) 55.00 MR0102806 (21() #1592) Kervaire,, Michel A. On the Pontryagin classes of certain ${\rm SO}(n)$-bundles over manifolds. Amer. J. Math. 80 1958 632--638. (Reviewer: W. S. Massey) 55.00 MR0094828 (20 #1337) Kervaire, Michel A. Sur les formules d'intégration de l'analyse vectorielle.
  • Surgery on Compact Manifolds, by C.T.C. Wall

    Surgery on Compact Manifolds, by C.T.C. Wall

    SURGERY ON COMPACT MANIFOLDS C. T. C. Wall Second Edition Edited by A. A. Ranicki ii Prof. C.T.C. Wall, F.R.S. Dept. of Mathematical Sciences University of Liverpool Liverpool L69 3BX England, UK Prof. A.A. Ranicki, F.R.S.E. Dept. of Mathematics and Statistics University of Edinburgh Edinburgh EH9 3JZ Scotland, UK Contents Forewords .................................................................ix Editor’sforewordtothesecondedition.....................................xi Introduction ..............................................................xv Part 0: Preliminaries Noteonconventions ........................................................2 0. Basichomotopynotions ....................................................3 1. Surgerybelowthemiddledimension ........................................8 1A. Appendix: applications ...................................................17 2. Simple Poincar´e complexes ................................................21 Part 1: The main theorem 3. Statementofresults .......................................................32 4. Animportantspecialcase .................................................39 5. Theeven-dimensionalcase ................................................44 6. Theodd-dimensionalcase .................................................57 7. The bounded odd-dimensional case . .......................................74 8. The bounded even-dimensional case .......................................82 9. Completionoftheproof ...................................................91 Part 2: Patterns