Quadratic Functions of Cocycles and Pin Structures

TL;DR

This paper establishes a bijective correspondence between Pin$^-$ structures on manifolds and certain quadratic functions on cocycles, extending classical results to include Pin$^-$ cases using advanced homotopy theory techniques.

Contribution

It introduces a new framework linking Pin$^-$ structures to quadratic functions on cocycles, generalizing previous results for Spin and surface cases with novel homotopy-theoretic methods.

Findings

Constructed a bijection between Pin$^-$ structures and quadratic functions.

Extended classical quadratic refinement results to Pin$^-$ manifolds.

Utilized stable homotopy theory of Postnikov towers for the extension.

Abstract

We construct a natural bijective correspondence between equivalence classes of Pin$^-$ structures on a compact simplicial $n$-manifold $M^n$, possibly with boundary, and $\mathbb{Z}/4$-valued 'quadratic functions' $Q$ defined on degree $n-1$ relative $\mathbb{Z}/2$ cocycles, $Q \colon Z^{n-1}(M^n, \partial M^n ; \mathbb{Z} /2) \to \mathbb{Z}/4$. The 'quadratic' property of $Q(p+q)$ 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 $H^1(M^n ; \mathbb{Z} /2)$. 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.