The Pontrjagin Dual of 4-Dimensional Spin Bordism

TL;DR

This paper investigates the Pontrjagin dual of 4-dimensional Spin bordism, describing it explicitly via cochain triples and their relations, extending previous work on 3-dimensional cases and motivated by physics applications.

Contribution

It provides a detailed description of the Pontrjagin dual of 4D Spin bordism as equivalence classes of cochain triples, generalizing earlier 3D results and applicable to classifying spaces of finite groups.

Findings

Explicit characterization of the dual group G(X) as cochain triples

Description of the pairing with 4D Spin manifolds

Extension of previous 3D Spin bordism duality results

Abstract

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.