Quadratic enhancements of surfaces: two vanishing results

Abstract:

This paper records two results which were inexplicably omitted from the paper on Pin structures on low dimensional manifolds in the LMS Lecture Note Series, volume 151, by Kirby and this author. Kirby declined to be listed as a coauthor of this paper.

A $Pin^{-}$-structure on a surface $X$ induces a quadratic enhancement of the mod $2$ intersection form, $q\colon H_1(X;\mathbb {Z}/2\mathbb {Z})\to \mathbb {Z}/4\mathbb {Z}$.

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 in their paper in Homology, Homotopy and Applications, volume 10. The arXiv version, arXiv:0707.1599 [math.AT], referred to an email from the author to Kreck for the proof. A more polished and public proof seems desirable.

In Section 6 of the paper with Kirby, a $Pin^{-}$-structure is constructed on a surface $X$ dual to $w_2$ in an oriented 4-manifold, $M^4$. Theorem 2.1 says that $q$ vanishes on the Poincaré dual to the image of $H^1(M;\mathbb {Z}/2\mathbb {Z})$ in $H^1(X;\mathbb {Z}/2\mathbb {Z})$.