Subject: Question about Kunneth formula for sheaf cohomology
Date: Mon, 21 Jan 2002 17:43:07 -0500 (EST)
From: Adam Sikora
To: Don Davis
Dear Don,
could you post the following question concerning Kunneth formula on your
list?
Thank you.
-- Adam Sikora
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.
The proof of Kunneth formula for singular cohomology relies on
the Eilenberg-Zilber theorem (C_*(X x Y) and C_*(X)\otimes C_*(Y) are
homotopic chain complexes).
The problem with sheaf cohomology is that the corresponding homology
(Borel-Moore homology) is very messy and
there is no obvious relation between C*(X x Y) and C*(X)\otimes C*(Y)
(except when X,Y are locally compact and compact supports are considered).
Therefore, I suspect that this may be a difficult question
showing that sometimes singular cohomology has better properties than
sheaf-Cech- Alexander-Spanier cohomology.
-- Adam Sikora