Subject: categories: duality question: alg. topology

From: Michael Barr <[email-masked]>
Date: Fri, 17 Jul 1998 19:16:20 -0400 (EDT)

To all experts in algebraic topology:

Is the duality I describe below known?  It does not appear to be
Poincare duality since there is no separate treatment of the torsion and
torsion free parts.  Let S be a simplicial complex, that is a set of
faces of an N simplex closed under face operations.  For what I am about
to say, it is necessary that the empty face in dimension -1 be included
so that the (co-)homology will be reduced.  This means, for example,
that there is a distinction between the empty simplicial complex (which
has no homology) and the simplicial complex consisting of only the empty
set (which has one cyclic homology group in degree -1).  In fact, the
former is dual to the N simplex and the latter to the N sphere.

Let S' be defined as the set of complements of the complement of S.
That is, for each simplex sigma not in S, the complementary simplex,
whose vertices are those not in sigma, belongs to S' (and nothing else).
Then for i = -1,...,N, the ith homology of S is the N-i-1st cohomology
of S'.

The algebraic topology books I have looked at do not mention Poincare
duality, except for Lefschetz, written in 1940, and he uses a complex
different from S'; in fact just opposite poset to S, since he is doing
things in the generality of an arbitrary poset equipped with "incidence
numbers" that are used in defining the boundary operator.  In
particular, his dual has the same number of elements as S, rather than
2^{N+1} - that number.

One thing I find curious is that although this gives, essentially, the
cohomology of S, there is no obvious way of making that into a
contravariant functor.