Subject: more on Poincare duality
From: James E McClure
Date: Thu, 12 Jul 2007 11:00:15 -0400 (EDT)
I'd like to clarify and correct my earlier posting about Poincare duality.
Peter is correct in saying that there are several different definitions of
the
cap product in the literature; I'll use Dold's definition because (unlike
all
the other textbooks I've consulted) he always pays careful attention to
the
standard (Koszul) sign convention about moving things past each other.
Let M be an m-dimensional compact closed manifold with fundamental
class [M]. Dold defines the Poincare duality map from H^*M to H_{m-*}M by
taking x to x \cap [M]. I am suggesting that a better definition takes
x to (-1)^{m|x|} x\cap [M] (in my earlier posting I assumed that this is
equal
to [M] \cap x, but that was a mistake). I have four reasons for saying
this is
better:
1) it is induced by a chain map (as I explained in my previous posting)
2) the umkehr map f_! (defined as the Poincare dual of f^*) has good
multiplicative properties. (See page 314 of Dold. Dold inserts a fudge
factor
in order to get good multiplicative properties; this factor arises
naturally from the version of Poincare duality I'm suggesting).
3) the umkehr map agrees with the Pontryagin-Thom construction. (See
Section
VIII.11 of Dold; he uses the same fudge factor to get this).
4) the Poincare dual of the cohomology exterior product can be written in
a
nice way.
All of these properties are needed for work that Greg Friedman, Scott
Wilson
and I are doing; Greg's forthcoming paper on the chain-level intersection
pairing for PL pseudomanifolds will include a careful account of 2) and
4).
I haven't found any place in the literature that gives this version of the
Poincare duality map. It would be interesting to know whether this
version is
consistent with Peter's theory.
Jim