Subject:
Re: four responses
From:
"Nicholas J. Kuhn" <njk4x@cms.mail.virginia.edu>
Date:
Mon, 14 Feb 2005 10:39:20 -0500
To:
Don Davis <dmd1@Lehigh.EDU>

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 <adem@math.ubc.ca>
> 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.
>
>
>