The Pontrjagin Dual of 3-Dimensional Spin Bordism
  For each space X we define an explicit group, G(X), functorially in X. This group is constructed from the groups of cochains on X. Furthermore, we construct an explicit functorial pairing with values in R/Z between the cochain representatives for elements of G(X) and maps of closed 3-dimensional spin manifolds to X. This pairing induces a pairing between G(X) and the 3-dimensional spin bordism group of X and identifies each with the Pontrjagin dual of the other. 
Greg Brumfiel and John Morgan
math.AT