Subject:Re: question abt sbgps of H^1
From:"Nicholas J. Kuhn" <njk4x@cms.mail.virginia.edu>
Date:Tue, 01 Feb 2005 11:18:56 -0500

Here is an answer to the question sent in by Dave Rusin:

With d(X) and n(X) as in the message (copied below),

(1) d(X) is at least n(X), and
(2) (d(X), n(x)) can be any pair subject to (1).

Proof of (1):  (This would make a good exercise for a graduate course.)  A surjection pi_1(X) --> F_n can be realized by a map f: X --> W(n)=wedge of n circles.  This maps splits, so f^* is monic, and the image of H1(W(n)) --> H1(X) is an isotropic submodule of dimension n.

Proof of (2):  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).

By construction, H1(V) --> H1(G(V)) is an isomorphism, and all products in H2(G(V)) are zero.  Let BG(V) be the classifying space for the finite group G(V).  Then n(BG(V))=0 (pi_1 is finite), but d(G(V)) = dimension of V.

The first example is BZ/4 with d=1, n=0.  The next example, with V=Z/2xZ/2 so BG(V) has d=2 and n=0, is the classifying space of a group of order 32 (#18 on Jon Carlson's website).

These examples can now be wedged with W(n)'s to create examples realizing any pair (d,n) with d at least n.

Remark:  G(V) is the universal 2-group with quotient V having all elements of order p central in G. (a `pC' group in Alex Adem's terminology).

Nick Kuhn
University of Virginia

--On Tuesday, February 01, 2005 7:00 AM -0500 Don Davis <dmd1@Lehigh.EDU> wrote:

> Subject: Isotropic subgroups of H1
> From: Dave Rusin <rusin@math.niu.edu>
> Date: Mon, 31 Jan 2005 11:22:03 -0600 (CST)
> One posting.........DMD
> __________________________________________________
>
> My colleague Dan Grubb sent me this question to post to topologists:
>
> Let X be a 'sufficiently nice' space, say at least locally
> path connected.
>
> An isotropic submodule M of H1 (X) is one where x,y \in M
> imply that x\cup y=0 in H2(X).  Let d(X) denote the maximal
> rank of an isotropic submodule of H1(X) (coefficients in the
> integers mod 2 although integers would be good for oriented
> manifolds).
>
> Now let n(X) denote the largest integar so that there is a
> surjection from pi_1 (X) to the free group on n(X) letters.
>
> How do d(X) and n(X) relate?
>
> I know that (with Z/(2) coefficients), they are equal for
> 2-dimensional closed manifolds. Also, if U and V are connected
> open sets which cover X, then the number of components
> of U\cap V is at most 1+max{d(X),n(X)}. Equality is preserved
> by joins, but I am not sure if it is by products.
>
> --Dan
>