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The Algebraic Surgery Exact Sequence

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The Algebraic Surgery Exact Sequence 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.
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