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Surgery with Coefficients PDF

164 Pages·1977·4.872 MB·English
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Lecture Notes ni Mathematics Edited yb .A Dold and .B Eckmann 195 G. A. Anderson Surgery with Coefficients I I Springer-Verlag Berlin. Heidelberq • New York 91 77 Author Gerald A. Anderson Department of Mathematics Pennsylvania State University University Park PA 16802/USA AMS Subject Classifications (1970): 57 B10, 57 C10, 57 D 65 ISBN 3-540-08250-6 Springer-Verlag Berlin • Heidelberg • New York 1SBN 0-387-08250-6 Springer-Verlag New York • Heidelberg • Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, -er printing, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, dna storage ni data banks. Under § 54 of the German Copyright Law where copies are made for other than private a use, eef is to payable the publisher, the amount of the fee to be with the publisher. agreement by determined © by Springer-Verlag Berlin - Heidelberg t977 Printed in Germany Printing dna binding: Beltz Offsetdruck, Hemsbach/Bergstr. 012345-0413/1412 INTRODUCTION This set of notes is derived from a seminar given at the University of Michigan in 1973, and portions of the author's doctoral thesis. It is intended to give a reasonably complete and self-contained account of surgery theory modulo a set of primes. The first three chapters contain the background material necessary to describe the theory. Chapter i is mainly definitions and notation and contains no new ideas, with the exception of relative localization and colocalization of spaces. Included is a sketch of the immersion classification theorem of Hirseh and Haefliger-Poenaru. Chapter 2 contains the theory of local Whitehead torsion. The definition differs from the one given by Cappell and Shaneson, but is justified by a Whitehead-type local collapse-expansion theorem. Chapter 3 discusses the theory of spaces which satisfy Poincare duality with coefficients in a ring, including the construction of a local Spivak normal fibration. Normal invariants modulo a set of primes are described and the homotopy groups of the classifying space Gp/H are computed. Chapter 4 contains the main surgery obstruction theorem. Briefly, groups are constructed to measure the obstruction to finding a homotopy equivalence (over a ring and with given torsion) cobordant to a given map. Below the middle dimension, the technique is due to Milnor and Wallace. Considering homotopy equivalences over the integers, ~he simply connected case is essentially done by Kervaire and Milnor, and globalized by Browder and Novikov; the general case is due to Wall. We show that the obstruction lies in a Wall group of a localized group ring. Surgery over a field was first considered by Petrie and Passman, and Miscenko noticed that Wall's groups behaved nicely away from the prime .2 More recently, Connelly, Giest and Pardon have considered rational surgery (in the non-simple case), and the methods of Cappell and Shaneson (which uses rings with a local epimorphism ~ + R) also apply. The general case, with rings of the form R~, is due to the author in his thesis. Chapter 5 gives the geometric definition of surgery groups, and the generalization to manifold n-ads. Quinn's approach is also briefly discussed. Finally, the periodicity theorem, in the non-simple case, is proved. Chapter 6 describes the result of changing rings in surgery groups by means of a long exact sequence. Corollaries include a Rothenberg-type sequence, the general periodicity isomorphism and determination of the kernel of s ~) s L2k_l( ÷ L2k_l(~) ~ finite, by simple linking forms, generalizing the original odd-dimensional surgery obstructions due to Wall and clarified by Connelly. Finally, five appendicies are included: Whitehead torsion notions for n-ads, the algebraic construction of Ln(W÷w';R) , the computation of Ln(~;R), surgery on embedded manifolds, and homotopy and homology spheres. The has reference been arranged into categories. Undoubtedly, some errors and omissions have occurred in this arrangement, but I hope the general drift is helpful to the reader. A number of people have been of greaD help in writing these notes. I am indebted to my thesis advisor C.N. Lee for many helpful suggestions and discussions. I would also llke to thank Dennis Barden, Allan Edmonds, Steve Ferry, and Steve Wilson, who participated in the seminar, Frank Raymond, Jack Mac Laughlin and W. Holstztynski. Massachusetts Institute of Technology TABLE OF CONTENTS Chapter i. Preliminaries i.i Modules ...................................... i 1.2 Homology and Cohomology with Twisted Coefficients ................................. 2 1.3 A-Sets ....................................... 4 1.4 Microbundles, Block Bundles and Spherical Fibrations ................................... 7 1.5 The Immersion Classification Theorem ......... ll 1.6 Intersection Numbers ......................... 14 1.7 Algebraic K-Theory ........................... 19 1.8 Localization ................................. 23 Chapter 2. Whitehead Torsion 28 Chapter .3 Poincare Complexes 39 3.1 Poincare Duality ............................. 39 3.2 Spherical Fibrations and Normal Maps ......... 45 Chapter 4 Surgery with Coefficients 54 4.1 Surgery ...................................... 54 4.2 The Problem of Surgery with Coefficients ..... 57 4.3 Surgery Obstruction Groups ................... 60 4.4 The Simply Connected Case .................... 74 4.5 The Exact Sequence of Surgery ................ 80 Chapter 5. Relative Surgery 82 5.1 Handle Subtraction and Applications .......... 82 5.2 Geometric Definition of Surgery Groups ....... 85 5.3 Classifying Spaces for Surgery ............... 94 5.4 The Periodicity Theorem, Part I .............. 96 Chapter 6. Relations Between Surgery Theories I01 6.1 The Long Exact Sequence of Surgery with Coefficients ................................ i01 6.2 The Rothenberg Sequence ..................... 105 6.3 The Periodicity Theorem, Part II ............ 109 6.4 Simple Linking Numbers ...................... Ii0 Appendix A. Torsion for n-ads 122 Appendix B. Higher L-Theories 124 Appendix C. L Groups of Free Abelian Groups 127 Appendix D. Ambient Surgery and Surgery Leaving a Submanifold Fixed 129 Appendix E: Homotopy and Homology Spheres ................. 135 References ................................................. 138 Symbol Index ............................................... 154 Index ...................................................... 156 Chapter i. Preliminaries i.i. Modules. Let A be a ring (not necessarily commutative) with involution, i.e. a map A ÷ A, written ~ k ~*, so that (a) (~i+12)* = ll* + ~2' (b) (Ii~2)* = ~2'~i* (c) ~** = ~. We will usually assume E i A. A denotes the group of units in A. Unless otherwise stated, all A-modules will be finitely generated and right. Let M be a A-module. Then M inherits a left A-module structure by defining l.m=m.l*. The dual of M is defined by M* = HomA(M,A) with A-module structure given by (f'~)(m) = l*f(m), f~ M*, l~ A. If M and N are A-modules, we define M ~N by giving N a left A-module structure as above. In the case N=A, M @AA is a A-module with (x @ l)W = x@ ~*l. A A-module is free if it is isomorphic to a direct sum of copies of A. M is projective if there is a A-module N so that M @ N is free. M is stabl[ free (s-free) if we may choose N to be free, i.e. M ~ k A is free for some k. If M is s-free, a stable basis (s-basis) is a basis for some M@A k. The main example of A will be a group ring, A = Rw for some (usually commutative) ring R with i, a multiplicative group with a homomorphism w:w ~ {±l}. The ring Rw is defined to be the set of all finite sums ~ng.g, ngg R, ge w. The involution is given by (~ng.g)* = ~w(g)ngg -1 1.2. Homology and Cohomology with Twi.sted Coefficients. Let X be a finite CW-complex, ~ = ~l(X) and w:~ ÷ {±i} a homomorphism. Let A = H~ and M a A-module. Define ~{i(x;M) = HI(C,(X) )~.~,,@ I} i(X;M) = H i(HomA(c ,(X),M)), where C,(X) is the chain complex of cellular chains in the universal cover X; C,(X) is a chain complex of free and based A-modules. If X is not compact, we use cohomology with compact supports. We write Hi(X) , Hi(x) for HI(X;A), Hi(X;A). We can define this alternately as follows: w determines an element in HI(x;~/2H) and so a double cover E ÷ X. Let Z/2H act on H non-trlvially and define t H to be the bundle associated to E with fiber H. Now ÷ X is a principal x-bundle and so define ~ to be the associated bundle with fiber M. Let ~t = ~ ~ zt. Then Hi(X;M) = Hi(X;~ t) Hi(x;M) = Hi(x;~) where we use bundle (or sheaf) homology and cohomology. If c is an n-cell in X, cap product defines en:cq(x) ÷ Cn_q(X) @ A ~ Cn_q(X) where cq(x) ~ HomA(Cq(X),A). This extends linearly to chains in Cn(X) and, in fact, to infinite chains since we are using compact supports. This defines ~N:Hq(X) + Hn_q(X) for ~E Hn(X). If M is a A-module, define ~:Hq(X;M) + Hn_q(X;M) by the composition HomA(Cq(X),M) ~ cq(x) @^M ÷ Cn_q(X)@^M. If f:X ÷ Y is a map, f#:Wl(X) ~ Wl(Y) , then define Ki(X;M ) = ker(f,:Hi(X;M ) ÷ Hi(Y;M)) Ki(X;M) ~ coker (f*:Hi(y;M) ÷ Hi(X;M)). The condition wI(X) ~ Wl(Y) isn't necessary, but will suffice for our purposes. A map f:X + Y with f#:Wl(X) ~ Wl(Y) is a homolo6~equivalence over R if f,:H,(X;Rw) + H,(Y;Rw) is an isomorphism. X and Y have the same R-homology tFpe if there is a sequence X = Z0,ZI,...,Z m = Y and homology equivalences over R~ i ÷ Z Zi+ I or Zi+ I ÷ Zi~ for each i.

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