Two responses to Goodwillie question............DMD
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Subject: Re: 2 questions
Date: Thu, 3 Oct 2002 17:16:55 +0200 (CEST)
From: Thomas Schick
>
> Subject: question for list
> Date: Wed, 2 Oct 2002 12:15:53 -0400 (EDT)
> From: Tom Goodwillie
>
> Is there a torsion-free discrete group G such that the
> integral cohomology ring of BG is isomorphic to that of
> infinite complex projective space?
>
> Tom Goodwillie
> ___________________________________________________
Baumslag, Dyer, Heller: the topology of discrete groups; give a refined
version of the Kan-Thurston
construction of groups with classifying spaces mapping by a homology
equivalence (all possible coefficients) to a given simplicial complex.
This works also for infinite simplicial complexes, and the construction
(iterated HNN-extensions and amalgamated products, starting with certain
basic building blocks) can be carried out in
such a way as not to create any torsion. Such a group should do the job.
Of course, in the case at hand this is an infinite construction, so there
is no need to hope that the group will be finitely generated.
Thoms Schick
>
-----------------------------------------------------------
Thomas Schick | email: schick@uni-math.gwdg.de
Bunsenstr. 3 | phone: ++49 551 397766
37073 Goettingen | fax: ++49 551 392985
Germany | http://www.uni-math.gwdg.de/schick
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Subject: H^*(BG) = H^*(CP^\infty)
Date: Thu, 3 Oct 2002 15:03:46 +0100 (BST)
From: Ian Leary
Yes there is. For any simplicial complex L, G. Baumslag, E Dyer
and A Heller construct a torsion-free group G_L and a map
BG_L ---> L inducing an isomorphism on cohomology (any local
coefficients on L). The construction is natural for injective
simplicial maps (eg. for automorphisms of L). Moreover, if L
is finite, then so is BG_L.
The reference is: The topology of discrete groups, JPAA 16 (1980),
pp. 1-47.
Somebody (maybe the same authors?) used this sort of construction
to show that any abelian group embeds as a central subgroup of an
acyclic group: given an abelian group A, take L to be an Eilenberg
-Mac Lane space K(A,2). The required group is the central extension
A >---> E --->> G_L
with the `same' extension class as the path-loop fibration over K(A,2).
Jonathan Cornick and I use this to exhibit non-Quillen groups (= groups
for which the Quillen map on mod-p cohomology is not an isomorphism):
one way around, take L = BC_p, then G_L is a torsion-free group whose
cohomology is the same as C_p. The other way around take L= K(C_p,2)
and make a group E as above. This E is an acyclic group containing
just one subgroup of order p. Our reference is: `Some remarks
concerning degree zero complete cohomology', Une degustation topologique:
Homotopy theory in the Swiss alps, Contemp. Math. 265 (2000) pp. 20-25.
Best wishes,
Ian Leary