By J. F. Davis
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Additional info for A Survey of Spherical Space Form Problem
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 . 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.
Then there is a unique maximal p-hyperelementary subgroup Hp for every prime p. 18 we can adjust any generator g E H" (0; Z) at the p-primary part only, to guarantee rHp(x(g» = 0 for all p. Then X(g) = 0 by Swan's induction theorem. Next suppose that 0 is not of type I, H m ( 0; Z) = Z/IOI, and that all 2-hyperelementary subgroups admit linear fixed-point free representations of dimension m. This is the case in (a) with m = 2n and in (b) with m = n. All periodic groups have a unique maximal p-hyperelementary subgroup Hp for all odd p.
At this point we note that since k* is primitive (2g )*( k*) == 0 in mod 2 cohomology, so the surgery problem corresponding to (2g) must have trivial surgery obstruction. Moreover after doing surgery the total space of the universal cover will be some homotopy sphere, and for these addition in the group of homotopy spheres r "-1 and addition of maps give the same result. Hence, since r" _I is finite, there is some finite multiple 2JL . 4 is necessary and sufficient to find an action of 'TT on S"-I.
A Survey of Spherical Space Form Problem by J. F. Davis