Quadratic Functions of Cocycles and Pin Structures

We construct a natural bijective correspondence between equivalence classes of Pin${}^{-}$ structures on a compact simplicial $n$-manifold $M^n$, possibly with boundary, and $\mathbb{Z}\mathrm{/}4$-valued ‘quadratic functions’ $Q$ defined on degree $n-1$ relative $\mathbb{Z}\mathrm{/}2$ cocycles, $Q \colon Z^{n-1}(M^n, \partial M^n ; $\mathbb{Z}$ /2) \to $\mathbb{Z}$/4$\mathrm{.}Th{e}^{\mathrm{\prime }}quadrati{c}^{\mathrm{\prime }}propertyof$Q(p+q)$ and the values $Q(dc)$ on coboundaries are expressed in terms of higher ${\cup }_i$ products of Steenrod. For $n=2$ the results extend old results relating Pin${}^{-}$ structures on closed surfaces to quadratic refinements of the cup product pairing on $H^1(M^n;\mathbb{Z}\mathrm{/}2)$. In the oriented case, that is, for Spin manifolds, the results extend results of Kapustin, see arXiv:1505.05856v2, and results in our previous paper on the Pontrjagin dual 4-dimensional Spin bordism, see arXiv:1803.08147. The extension of those results to Pin${}^{-}$ manifolds in this paper required a different approach, involving some stable homotopy theory of Postnikov towers.