  The goal of this paper is to study the Pontrjagin dual of (reduced)
4-dimensional Spin bordism. That is to say, we consider the functor from the
category of topological spaces to the category of compact abelian groups that
associates to each space X the compact group of homomorphisms from the reduced
4-dimensional Spin bordism of X to the circle. In a previous paper, we studied
the analogous problem for 3-dimensional Spin bordism. Our work was motivated by
some questions from physics. The physicists are primarily interested in the
case when X is the classifying space of a finite group, but our arguments are
valid for general X. We describe the dual group, G(X), as equivalence classes
of triples of cochains (w,p,a) on X, triples satisfying certain relations with
a product. We also describe the pairing between such triples and a closed
4-dimensional Spin manifold mapping to X, the pairing that produces the
identification of G(X) with the Pontrjagin dual of the reduced 4-dimensional
Spin bordism of X.
