C2 of the cellular chain complex Cfr(M, E) is given by the (1 x I)-matrix i-I where i = pr(t) E G. By a direct computation, Tpr(M, E,w') = -(I - 1)-1 up to multiplication by powers of i. 4 implies that (with the same ambiguity) Tpr(E, w) = -(I - 1) Tpr(M, e, WM) = -(I - 1) pr(T(M, e)).
Z' is additive in z and therefore depends only on z(mod r) E H/r. 1. Lemma. There is a unique square volume form pseudo-basis h = (hI, ... , h n ) of H(p) , If L is non-degenerate, then so is O~. or, on H / r such that for any 44 Cbapter III. TbeTorsion versus tbe Cobomology Rings Proof. Each pseudo-basis h = (hI, ... , h n ) of H(p) gives rise to a square volume form Oh on Hlr as follows: for a basis a of Hlr, set Oh(a) = [alhj2 det (hi' hj)i,j=l, ... ,n' It is clear that Oh(h) = det (hi' hjkj and we need only to show that Or, = Oh does not depend on h.