Quadratic enhancements of surfaces: two vanishing results

TL;DR

This paper presents two mathematical results about quadratic enhancements induced by Pin^- structures on surfaces, showing their vanishing on specific homology kernels and duals, with implications for low-dimensional topology.

Contribution

It provides two new vanishing theorems related to Pin^- structures on surfaces, filling gaps in previous literature and offering clearer proofs for these properties.

Findings

q vanishes on the kernel of the homology map to a bounding 3-manifold

q vanishes on the Poincare dual to the image of H^1(M^4;Z/2Z) in H^1(F;Z/2Z)

Clarifies properties of quadratic enhancements induced by Pin^- structures

Abstract

This note records two results which were inexplicably omitted from our paper on Pin structures on low dimensional manifolds, [KT]. Kirby chose not to be listed as a coauthor. A Pin^- structure on a surface F induces a quadratic enhancement of the mod 2 intersection form, q: H_1(F;Z/2Z) -> Z/4Z Theorem 1.1 says that q vanishes on the kernel of the map in homology to a bounding 3-manifold. This is used by Kreck and Puppe (arXiv:0707.1599 [math.AT]) who refer for a proof to an email of the author to Kreck. A more polished and public proof seems desirable. In [KT], section 6, a Pin^- structure is constructed on a surface F dual to w_2 in an oriented 4-manifold M^4. Theorem 2.1 says that q vanishes on the Poincare dual to the image of H^1(M^4;Z/2Z) in H^1(F;Z/2Z).