By J. F. Davis
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Extra info for A Survey of Spherical Space Form Problem
43 nr:GL(zC~I)G)~YU+(Ki). 30 thus reduces to the calculation of units in cyclotomic number fields. The problem of determining totally positive units seems difficult and interesting. For some results, see . 7 we see that if the Swan obstruction a n ( G) vanishes then an(H) = 0 for every p-hyperelementary subgroup H. Of course, a priori this last condition for rn(t) may not be sufficient. It does suffice however, to find agE H n (G; Z) so that rH (X(g)) = 0 for all p-hyperelemen- tary subgroups.
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 . The following theorem is an extension of ideas from . 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. 11 Let K/Q be totally real and Galois of degree 2'.
B. 1. 1) an element. 9 g( C) E Ext~(L, M). 10 (K. Roggenkamp-C. T. C. 8) with the Pifinitely generated projective if and only iffor any A-module N it is the case that u g: Ext~(M, N) ~ Ext~+i(L, N) is an isomorphism for i> 0 and an epimorphism for i = o. 240 1. F. DAVIS AND R. J. 12 (Wall) The Euler characteristic x( C) depends only on g E Ext~(L, M) and not on the particular sequence which represents it. 13 O~Z~Pn-I~"'~PJ~PO~Z~O with the Pj finitely generated projectives. Conversely, if there exists a sequence of the above type for an arbitrary group G, then H n (G; Z) = ZI GI so that G is a P-group of period n.
A Survey of Spherical Space Form Problem by J. F. Davis