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**

**Example text**

Are matrix rings M (

So T(a) -a 52 §8. The Local Lifting Theorem We recall from [C-M] the definition of the boundary map Given a Hermitian space over we choose a /\ . /L. Let a pairing £1 /~free submodule /\,(H,S) , where the module and S £2 M x M --+ /\sjl\ is defined by are representatives for m l S(m ,m 2 ) l and m 2 S(£1'£2) , where = if! in The torsion Hermitian space (M,S) , which is of the same symmetry as defined to be the image under We denotes of the class of (H,S) specialize to the case /\ = ~ (pPT),1\ localized at the prime ~ p ,and 7T involution on both these rings is specified by s Now, (H,S) , is now in q:)(7T) ,where ~ is a finite group.

Tive or negative type respectively. We say that two involutions there is an automorphism means that (M (D), T) n a and of T and of so that Mn(D) Mn(D) T • a are equivalent if a 0 a. 3: l i A' = BAr (B) ,where then TA Proof: is equivalent to Define TA 0 a(M) -1 = is a non-singular matrix over D, T , . A B MB a(M) B AT(B -1 Then we have -1 MB)A 1 1 T on Mn(D) in CD up to equivalence AT(B)T(M)T(B- )A- = B-1A'T(M)A,-lB Remark: FT Thus, the classification of involutions equal to a given subfield of index 1 or 2 with is equivalent to the classification of non-singular E-a-symmetric matrices A under the equivalence relation 0' some non-singular matrix a is a particular involution with Fa F.