By J. F. Davis

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This is accomplished by means of pullback diagrams and Mayer-Vietoris type sequences in K-theory. Another philosophy is the Hasse principle-to solve a problem over Z, solve it first locally at all primes, and then globally. 29 Here Zp = lim Z/ pn is the p-adic completion of Z and Qp is its quotient field. For the arithmetic and geometric properties of these rings see [21], [11]. 30 ;, U Kt(QpG) ~ Ko(ZG) pllGI ~ K o( zC ~I) G) Et) PrJ,1 Ko(ZpG) ~ U Ko(QpG). pllGI PERIODIC RESOLUTIONS 247 See Milnor [29] for the definition of the boundary map.

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'. (a) E+(K) = E 2 (K) if and only if there is a u E E(K) such that, Nu = -1. ) (b) Let H be a subgroup of E (K) offinite index. If there is a u EH with Nu = -1, then IE(K)/ HI is odd. THEOREM Let An = 'n +,~I. 12 (~) = -1, If K is Kp' or K pq with the quadratic or the maximal 2 -extension of Q in Q[A 2 ", Apl with p ~ 1 (mod 8) then E 2 (K) = E+(K) and the cyclotomic units have odd index in U(K).

Hence we get lifting here. For p = 2, TT"p = Z/2) for which we use the previous argument or TT"p = Q(2 n • 1,1,1). But here (Z/2 n)"/squares = Z/2 xZ/2 with generators -I, 5 and since -1 doesn't matter, just 5 is important. On the other hand T( Q(2 n I, I» = Z/2, has generator (5), [16] and thus there is a unique homotopy type which contains a finite complex in dimensions 8k + 3. I)], and so the set of manifolds is precisely the set of finite homotopy types, so once more we get reduction. Now L I(Xn-1-pt)= 271 vV p J.