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)