By J. F. Davis

ISBN-10: 3718602504

ISBN-13: 9783718602506

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**Extra info for A Survey of Spherical Space Form Problem**

**Example text**

10 Let Kw c Q(~w) be the maximal totally real is a power of 2. 2 n abc. Computations are simplified greatly when the cyclotomic units are of odd index in the units or when all totally positive units are squares. For a reference on cyclotomic number theory and the definition of cyclotomic unit, see [63]. The following theorem is an extension of ideas from [33]. Let E (K) denote the units in the ring of integers of a number field K. Let E 2 (K) denote the squares in E(K). Let E+(K) denote the elements of E(K) which are positive at all real places.

Choose the integers b, b l , ••• , b r such that the map is degree 1 and the bi are even. 28 is covered by a map of tangent bundles, hence of stable normal bundles. 26 and the fact that L~n+,(F27T/rad) has exponent 2. 0 SECTION 5 The Surgery ProblemsGeneral Structure Let TT" be a P- group and suppose ITn ( TT") = 0, so there is a finite complex X n - I with fundamental group TT" and universal cover X n- I =Sn-l. e. 1 f:X ~Bsc where Bsc is the classifying space for (oriented) fiber homotopy spherical fibrations.

In the last case with p == 5 (mod 8), N K / Q N Q [A 2 ", Ap]/ K (A p - A2 ,,) = -1. If p == 3 or 7 (mod 8) then N K / Q ( -1- A2 ,,) = -1. This last unit gives Weber's result that E +( Q[A 2 ,,]) = E 2 ( Q[A 2 ,,]) and that h+(Q[A 2 ,,]) is odd. J. F. DAVIS AND R. J. 13 Let G be a p-group, Fp the finite field with p elements, and E : F pG ~ Fp the augmentation. If E (u) '" 0 then (FpGu) = FpG. LEMMA Proof Equivalently we show I ( G) = ker( E ) is the unique maximal ideal of F pG. If J is a maximal ideal then FpG/ J is irreducible, hence FpG/ J = Fp (see Serre [43]).

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