Read e-book online A Survey of Spherical Space Form Problem PDF

By J. F. Davis

ISBN-10: 3718602504

ISBN-13: 9783718602506

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T. C. Wall defined his finiteness obstruction. We now review some of the major results of these papers. Let C = {Cj , a} be a chain complex of projective Z7Tmodules, with 7T any group. 23 C is finitely dominated if C is Z7T-chain homotopic to a finite projective complex {Pi. ah",j",o. e. ) C is homotopy finite if C is Z7T-chain homotopic to a finite free complex. 24 Suppose C is Z7T-chain homotopic to a finite projective complex {Pj, a}k",j",O' Then depends only on the chain homotopy type of C.

The structure of im K,(Z(l/IGi)G) is more difficult to describe. We shall do so however since this is the source of the more delicate number theory involved in the calculation of the Swan obstruction for the groups Q(2 n a, b, c). Let QG = - i Ai be the decomposition of QG into a sum of simple algebras Ai. Let K i = center Ai and Ri = K i n Z(l/IGI)G. Note that Z( I11 GI)G contains all the central idempotents of QG so that the center of Z(l/IGI)G is - i Ri, where Ri is the integral closure of Z(l/IGi) in K i • A summand Ai is symplectic if R ®Q Ai is a sum of matrix rings over the real quaternions.

T [(Zip') is the augmentation ideal, the kernel of the augmentation E: Z(z/p') -'> Z defined by (L, n,g,) = L, n,. 254 1. F. DAYIS AND R. 1. MILGRAM TP'-I). Then set R equal to Now consider the pull-back diagram where the gluing automorphisms are the identity on Qq ® C and Qp ® V® C, while at Qp ® R the ReidemeisterDeRham torsions T of the gluing automorphisms are I at Qp(~pJ), j2:: 1 and pp' at the trivial representation Qp. But this torsion T satisfies a( T) = X( C q ). On the other hand, T is pS[p x- q1m Iq] at the trivial representation, and 1 at all the other representations.

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A Survey of Spherical Space Form Problem by J. F. Davis

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