Subject: item for the discussion list
From: James E McClure
Date: Sat, 7 Jul 2007 10:39:29 -0400 (EDT)
I spent some time recently untangling a confusion about Poincare duality
and I
thought it might be helpful for other people to know about it.
There are two ways to define the Poincare duality map from S^*M to S_*M:
either take x to o\cap x or to x \cap o (where o is a cycle representing
the
fundamental class). All the textbooks I've looked at (except for
Ranicki's
book Algebraic and Geometric Surgery) use the second choice, but it turns
out that the first choice is better. For example, if we convert S^*M to a
chain complex T_*M in the usual way (T_pM=S^{-p}M) then the first choice
for
Poincare duality gives a chain map (raising degrees by dim M) from T_*M to
S_*M but the second does not (it {\it commutes} with the differential,
which
is not what one wants for a chain map raising degrees by dim M). Dold's
book
Algebraic Topology uses the wrong choice, and Dold then observes that
there is
an anomaly in the definition of the umkehr map (page 314 and the top of
page
315); this anomaly would not have occurred if he had made the right choice
for
Poincare duality.