Subject: Isotropic subgroups of H^1
From: Dave Rusin <rusin@math.niu.edu>
Date: Mon, 31 Jan 2005 11:22:03 -0600 (CST)
One posting.........DMD
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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