Subject: Re: Hopf postings & refinement Date: Tue, 30 Apr 2002 12:50:50 -0500 From: Frank Connolly To: Don Davis An Answer to Stasheff's Question: The rational group ring, Q[Z/2Z] splits as Q^{+} X Q^{-}. By (twisted) Poincare duality, H^k(Y;Q^{-}) = H_{n-k)(Y;Q). We therefore get: H^k(X;Q)= H^k(Y; Q[Z/2Z])= H^k(Y;Q) X H^{k}(Y;Q^{-})=H^k(Y;Q) X H_n-k(Y;Q), from which: b_k(X) = b_k(Y) + b_{n-k}(Y) follows. -- Frank Connolly Professor Department of Mathematics Room 169 Hurley University of Notre Dame Notre Dame IN, 46556-4618 (219)631-6571