Subject: Re: Hopf postings & refinement Date: Tue, 30 Apr 2002 12:17:29 -0400 (EDT) From: Tom Goodwillie To: Don Davis > > Suppose Y is a non-orientable closed n-manifold and X is its orientation > > cover. > Is the relation I stated correct? > > b_k(X) = b_k(Y) + b_{n-k}(Y) > Yes. Use the deck transformation to split up the (rational, singular) chain complex of X into plus and minus eigenspaces. Plus part of H_k(X) = homology of plus part of chains on X H_k(Y) Minus part of H_k(X) = Plus part of H^{n-k}(X) by Poincare duality = Dual of plus part of H_{n-k}(X) = Dual of H_{n-k}(Y)