2008/02/01 by Laurence R. Taylor
Mathematics · #math.AT #math.GT
published as Proc. Amer. Math. Soc. 137 (2009), 1135-1138 · 3 pages
arxiv created 2008/02/01 · arxiv updated 2015/06/08
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: H1(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 w2 in an oriented 4-manifold M4. Theorem 2.1 says that q vanishes on the Poincare dual to the image of H1(M4;Z/2Z) in H1(F;Z/2Z).