Two responses to Goodwillie question............DMD ________________________________________________ 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 ----------------------------------------------------------- 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