Subject: a comment From: Alejandro Adem Date: Mon, 14 Feb 2005 17:33:42 -0600 (CST) To: Don Davis A comment on Nick's reply: if H is a proper subgroup of G(n), then it cannot map onto H_1(G(n), Z/2) = V(n) (it's a Frattini extension). So his assertion is correct. In the Galois cohomology world, the universal group G(n) corresponds to fields of cohomological dimension 1. More generally, the subring generated by 1-dim classes in the mod 2 cohomology of the W-group of F is the mod 2 Galois cohomology of the field. Alejandro Adem. >> _______________________________________________________________ >> >> Subject: Re: four responses >> From: "Nicholas J. Kuhn" >> Date: Mon, 14 Feb 2005 10:39:20 -0500 >> >> 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. >> > >> > >> > >> >> >> >> >>