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By G. Carlsson, R. James Milgram (auth.), Peter Hoffman, Victor Snaith (eds.)

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Extra info for Algebraic Topology Waterloo 1978: Proceedings of a Conference Sponsored by the Canadian Mathematical Society, NSERC (Canada), and the University of Waterloo, June 1978

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5: Let, or x. f : Proof: > 1 K Then F K < a,b > U A = -A U , u -h(U) so -A(U + LeU»~. < a,S> ® M (k) ® K 2 M (K) 2 K . 5 follows. = -y K < a, b > E k and and = k k < -< A,U >C k A, U + TU > 18 K . k < a,b>. 46 §7. Involutions and Division Algebras One of the terms in our exact sequence is is given an involution where each Di T by T(g) g = -1 is a division algebra over to a central idempotent in e. 1. ), n i 1. ~. Each summand corresponds I. ) n 1. i Moreover, as was pointed out to us by Proof: d i TITT We write 1 corresponding to the i-th representation.

II and , Fix('r' ) he the fjxed sets T of + , two type T' (x) = s (The sign can be chosen have type I(B), then we can assume if and only if usual involution on T(S) = S , then T, T' II) , then is of type T, if T, T , -1 I(B) T(X)S T has , then s. 9: Let D lie over k x = ki. < a,b > T'(XS)=s -1 But Nowassume Fix(T) = k(l) + k(j) +k(ij) (ki)-lT(X) (ki) is then the T'(X) = s-lT(x)s , and s'("l()s=xs i f and only i f x E Fix(T) and To complete the proof it suffices to note in the case of a type involution that if (as) -IT (as) T(S) = -s then there is an s- 1 1(X)S while a E k 1(as) = as .

Are all of odd order. l(iii) then Finally, for Z/2i, n(Z/ i) 2 2 neVi) 4 and and Z/2 . l(i) algebras. Note first that the involution on each of these is the usual involution, and transferring up does not change this. » 2 p 1. hence away from 2 these contribute nothing. (D anything. 1 (ii) algebras. 10: 52 Proof: parts at d (5) i-2 > , ••• etc. < E5A{> u, E units, are in the image which are either units or have norm 2. either case they cancel out. only have Ei > 5\ > < 5u Now, all of these but + Lo(~(Ai» < Now away from 2 all these remaining elements 5 is prime in _ 1(2 i ) , but Z(\) 52 i-3 $ 1(2 i ) Hence (5) splits in into 2 primes interchanged by complex conjugation.

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