Two responses to Hatcher's question about homology and homotopy..DMD _________________________________________ Subject: Re:Hatcher question Date: Thu, 08 Mar 2001 17:43:55 -0500 From: William Browder Use the Cartan-Leray spectral sequence, with E_2 = H*(G; H*U) where U= the unversal covering space of X, G = the fundamental group, which converges to the associated graded group to H*(X). Since H*(U) is finite in each dimension, and G is finite, it follows that E_2 is finite in each dimension, so H*(X) is finite. __________________________________________ Subject: Re: request and question Date: Fri, 9 Mar 2001 03:31:06 +0000 From: Tom Goodwillie >Question: Does anyone know an example of a space with finite homotopy >groups but some homology group infinite? (Not H_0 of course!) It can't happen. In fact, more generally if pi_1(X)=G is finite and every other homotopy group of X is finitely generated then, modulo the Serre class of finite abelian groups, the homology of X is the same as the G-(co-)invariant part of the homology of the universal cover E. (And the homology groups of E are finite if the homotopy groups are.) Proof: In the Serre spectral sequence of E -> X -> K(G,1) the group E^2_{i,j}=H_i(G;H_j(E)) is finite if i>0. Tom Goodwillie