Two responses to Hatcher's question about homology and homotopy..DMD
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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.
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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