Subject: Re: four responses From: "Nicholas J. Kuhn" Date: Mon, 14 Feb 2005 10:39:20 -0500 To: Don Davis Alex Adem's comments spur me to mention one more comment about the question posted before. Recall that d(X) was defined to be the largest d so that H1(X;Z/2) coeffiecients has a d dimensional subspace with all products 0 in H2. The original question asked for a comparison with n(X) = largest n so that there exists an epimorphism from pi_1(X) to F_n, the free group on n generators. Here is a characterization of d(X) in terms of the fundamental group of X. Let V(n) = (Z/2)^n, and let G(n) be the finite 2-group I described as a central extension H_2(V(n);Z/2) --> G(n) --> V(n). Proposition. TFAE (a) d(X) is at least n. (b) There exists a homomorphism from pi_1(X) to G(n) such that the composite pi_1(X) --> G(n) --> V(n) is epic. Related to (b), I am pretty sure that the following holds: ?? A homomorphism from H to G(n) is epic iff the composite H -> G(n) -> V(n) is epic. It is obvious that both F_n and (Z/4)*..*(Z/4) (n times) both admit homomorphisms to G(n) satisfying condition (b), proving that d(X) is at least as big as n(X), and fitting with the wedge of mod 4 Moore spaces example. Nick > Subject: comment on N.Kuhn remark > From: Alejandro Adem > Date: Fri, 11 Feb 2005 14:12:42 -0800 (PST) > > This comment is about a construction Nick Kuhn recently mentioned: > (see www.lehigh.edu/~dmd1/nk21.txt) > ------------------------------ > Let V be a finite dimensional Z/2 vector space (and let > homology and cohomology have Z/2 coefficients). There is a `universal' > central group extension H_2(V) --> G(V) --> V whose d_2 differential in > the Serre spectral sequence for H^*(G(V)) is an isomorphism from > H2(V)=E_2^{0,1} to E_2^{2,0}=H2(V). > -------------- > > The groups described by Nick are known as universal W-groups, and have > been > studied by Minac and Spira in applications to Galois cohomology. > Calculating > their mod 2 cohomology is an interesting open problem (see > math.AT/9812169). > > The mod p analogues have also been studied; their cohomology can be > completely > computed if p is large enough with respect to the dimension of V (see > math.AT/0008229). > > And yes, they are pC groups. > > Alejandro Adem. > > >