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Lecture Notes in Mathematics Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann 914 Max L. Warshauer The Witt Group of Degree k Maps and Asymmetric Inner Product Spaces Springer-Verlag Berlin Heidelberg New York 1982 Author Max L.Warshauer Department of Mathematics, Southwest Texas State University San Marcos, TX ?8666, USA AMS Subject Classifications (1980): 10 C 05 ISBN 3-540-11201-4 Springer-Verlag Berlin Heidelberg NewYork ISBN 0-387-11201-4 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, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under w 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich. by Springer-Verlag Berlin Heidelberg 1982 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210 TABLE OF CONTENTS INTRODUCTION ................................................ 1 CONVENTIONS ........................................ ~ .......... II Chapter I THE WITT RING ...................................... 12 i. Setting and notation .............................. 12 2. Inner products .................................... 14 3. Constructing new inner products out of old ........ 23 4. The symmetry operator ............................... 26 5. The Witt equivalence relation ..................... 33 6. Anisotropic representatives ....................... 41 Chapter II WITT INVARIANTS ................................. 47 i. Prime ideals ....................................... 48 2. Hilbert symbols .................................... 51 3. Rank .............................................. 54 4. Diagonalization and the discriminant .............. 55 5. Signatures ......................................... 66 Chapter III POLYNOMIALS .................................... 70 Chapter IV WITT GROUP OF A FIELD .......................... 79 i. Decomposition by characteristic polynomial ........ 79 2. The trace lemma ................................... 90 3. Computing Witt groups ................................ 100 4. Torsion in W(-k,F) ................................ 105 Chapter V THE SQUARING MAP .................................. 108 I. Scharlau~s transfer ................................ !ii 2. The exact octagon over a field .................... 123 IV Chapter VI THE BOUNDARY ~..~ ...................... , ...... ..... 133 i. The boundary homomorphism ........................ 133 2. Reducing to the maximal order ..................... 142 3. Computing the local boundary ~(D,P) for the maximal order ..................................... 152 4. Computing the cokernel of ~(D} ................... 167 Chapter VII NON MAXIMAL ORDERS ........................... 184 i. Traces and canonical localizers . .................. 184 2. Normal extensions ................................ 192 3. Computing trace for finite fields ................ 198 4. The conductor and T(M~ ........................... 203 Chapter VIII THE GLOBAL BOUNDARY ......................... 207 i. The coupling invariants ........................ ~. 207 2. The boundary is onto ............................. 210 Chapter IX A DETAILED ANALYSIS OF THE OCTAGON ............ 226 i. The involutions .................................. 226 2. The map I : We(k2,F) § We(-k,F) .............. 229 e 3. The map d s : We(-k,F) § A(F) .................. 238 4. The map Se: We(k,F) § We(k2,F) ............... 241 5. The map me: A(F) + We(k,F) ................... 242 Chapter X THE OCTAGON OVER Z ............................. 249 NOTATION .................................................. 259 REFERENCES ................................................ 264 INDEX ..................................................... 267 INTRODUCTION In these notes our goal has been to develop the algebraic machinery for the study of the Witt group W(k,I) of degree k mapping structures and the Witt group of asymmetric inner product spaces. We are particularly interested in their relationship which arises in an exact octagon which is studied both for a field F and for the integers Z We show that this octagon is the appropriate generalization of the Scharlau transfer sequence [!~n 201]. We have tried to develop the properties of these Witt groups in a self-contained and complete manner in order to make this accessible to a larger audience, When references are given they generally are specific including page numbers. However, we should mention two general references. First, The Algebraic Theory of Quadratic Forms by T.Y. Lam [Lm] develops the Witt group over a field of characteristic not equal to 2, Second, Note____~so_~n th___eeW itt Classification of Hermitian Innerproduct Spaces over a Ring of Algebraic Integers by P.E. Conner [C] discusses Hermitian forms and the Witt group over an algebraic number field and ring of integers therein. Together these should provide any background material the reader might need. Although the viewpoint we take is entirely algebraic, we would be remiss if we did not mention its topological motivation. Much of this work originated in our efforts to explain and exposit the important work of N.W. Stoltzfus, Unravellin~ the Integral Knot Concordance Group [Sf-l]. We shall describe this topological connection momentarily. First however we need to describe the algebraic objects at hand. Our object is to define and study a Witt group W(k,I) of degree k maps, where k is an integer, D is the underlying ring, and I is a D-module. This group consists of Witt equivalence classes of triples (M,B,/) satisfying: (i) B: M x M + I is an I-valued inner product defined on the D-module M. (2) /:M § M is a D-module homomorphism of M satisfying B(/x,/y) = kB(x,y). We refer to 1 as a map of degree k. A triple (M,B,/) is metabolic (Witt equivalent to zero) if there is an /-invariant submodule NC M with N = N ~. Here N ~ is the orthogonal complement. This enables us to define the Witt equivalence relation ~ by: (M,B,/) ~ (MI,BI,/I) iff (M ~ MI, B ~9 -B I, Z ~) /I ) is metabolic. The witt equivalence class of (M,B,/) is denoted [M,B,/]. The operation of direct sum ~ makes this collection of equivalence classes into a group W(k,I). The identity is the Witt equivalence class of metabolic triples. We also develop the basic properties of asymmetric inner product spaces (M,B) where no symmetry requirement is placed on the inner product B. The key to understanding these asymmetric inner products is a "symmetry operator" s satisfying B(x,y) = B(y,sx) for all x,y e M. As above we define the notion of metabolic by (M,B) is metabolic if there is an s-invariant subspace N C M with N = N ~. This leads us to define Witt equivalence as before, and there results the asymmetric Witt group. There is a very interesting relationship between this asymmetric Witt group and the Witt group of degree k maps. This comes from the squaring map S: W(k,D) + W(k2,D) given by [M,B,/] ~ [M,B,/2]. For D = F a field an eight term exact sequence is developed from S. This octagon involves the Witt group of asymmetric inner products A(F) just described. As a special case of the octagon we obtain the transfer sequence of Scharlau, Elman and Lain ~m 20~ and ~,L 23-2~. This appears as an exact octagon in which several terms vanish. We are able to reinter- pret the kernel and cokernel of this transfer sequence. Thus, these Witt group W(k,F) and A(F) are an appropriate generalization of the classical Witt group W(F), at least in so far as relating to and explai- ning the Scharlau transfer sequence. We should like to prove exactness in the octagon over Z. in order to continue we have a boundary sequence which relates W(k,Z) to W(k,Q) O ~ W(k,Z) + W(k,Q) ~ W(k,Q/Z) When k = • this sequence is short exact. Using this we are able to prove exactness in the octagon over Z when k = • by comparing the octagon to the octagon over Q. AS we have said, in these notes we are developing the algebraic machinery to study the Witt group of degree k maps and the asymmetric Witt group. We should discuss the topological motivation for this work. When k = +i the Witt group W(+I,Z) is the crucial element determining the bordism groups A n of orientation preserving diffeomorphisms of n-dimensional closed oriented smooth manifolds. Medrano introduced the appropriate Witt invariant as follows. Let f: M 2n + M 2n be an orientation preserving diffeomorphism of a closed, oriented, smooth 2n-dimensional manifold M 2n. We consider then the pair (M2n,f) in A2n. The degree +i mapping triple associated to this pair is (Hn (M2n;z)/torsion , B, f*) where B(x,y) = e. ((x ~ y) ~ [M2n]) , E, is augmentation, [M 2n] is the fundamental class, ~ is cup product, ~ is cap product, f* is the induced map on cohomology. If (M2n,f) bounds, this triple is metabolic. Thus there is an induced homomorphism I: A2n + W(+I,Z). The task of computing the bordism groups A2n. was completed by Kreck [K] who showed that this Witt invariant was essentially the only invariant for bordism of diffeomorphisms. More generally, given a closed oriented 2n-dimensional manifold together with a map 1 of degree k, the corresponding Witt triple (M,B,I) as above satisfies B(s = kB(x,y) We are thus led to examine the Witt group W(k,Z). The Witt group of asymmetric inner product spaces arises in Quinn's work [Q] on open book decomposition. The relation of this to the Witt group arising in Kreck's
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