Subject: Re: question
Date: Tue, 22 Jan 2002 20:07:00 +0100 (CET)
From: Marek Golasinski
You wrote:
============
Let H* be singular cohomology with coefficients in Z (integers).
For any topological spaces X, Y such that H^n(X) is finitely generated
for each n,
we have the following short exact sequence
0-> [H*(X)\otimes H*(Y)]^n-> H^n(X x Y)-> [H*(X) * H*(Y)]^{n-1}->0
which splits. (This is the Kunneth formula).
Question: Is this statement true for sheaf cohomology with (trivial)
coefficients in Z? Assume that X,Y are paracompact and hence H* is
isomorphic to Alexander-Spanier and Cech cohomologies.
=========
I think the question is related with the paper by me and
Daciberg Lima Goncalves, "Generalized Eilenberg-Zilber
type theorem and its equivariant applications",
Bull.Sci.Math. (1999) 123, 285-298.
Yours sincerely,
M.Golasinski
N. Copernicus University,
Torun, Poland
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Subject: Sheaf - Cech cohomology
Date: Tue, 22 Jan 2002 10:49:51 -0500 (EST)
From: Claude Schochet
If you are willing to use compact metric spaces then there is a very nice
homology theory dual to Cech cohomology - it is called Steenrod homology
and (remarkably enough) was developed by Steenrod in his 1940 paper in
Mich Math. J. It has a wedge axiom - the homology of the strong wedge is
the product of the homology of the individual pieces, and there is a UCT
with Cech cohomology as well as a lim - lim^1 sequence for the inverse
limit of finite complexes. Milnor wrote about this theory about 35 years
ago and his paper was published fairly recently (no - it was not sitting
on an editor's desk for that long.)
There is some work by some Russian topologists over the years re extending
this to more general topological spaces - I don't know the details.
One can axiomatize the theory, drop the dimension axiom, and construct
generalized Steenrod homology theories associated to any cohomology theory
- Kahn, Kaminker, and I did this years ago on the way to proving that the
Kasparov group KK_*(C(X),C) was "Steenrod K-homology."