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Quadratic enhancements of surfaces: two vanishing results

Author: Laurence R. Taylor
Journal: Proc. Amer. Math. Soc. 137 (2009), 1135-1138
MSC (2000): Primary 57R15; Secondary 57M25, 57R90
Published electronically: October 22, 2008
MathSciNet review: 2457455
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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})$.

References [Enhancements On Off] (What's this?)

  • R. C. Kirby and L. R. Taylor, $ {\rm Pin}$ structures on low-dimensional manifolds, Geometry of low-dimensional manifolds, 2 (Durham, 1989) London Math. Soc. Lecture Note Ser., vol. 151, Cambridge Univ. Press, Cambridge, 1990, pp. 177-242. MR 1171915 (94b:57031) Matthias Kreck and Volker Puppe, Involutions on $ 3$-manifolds and self-dual, binary codes, Homology, Homotopy Appl. 10 (2008), no. 2, 139-148 (electronic).

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Additional Information

Laurence R. Taylor
Affiliation: Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556

Received by editor(s): February 1, 2008
Published electronically: October 22, 2008
Communicated by: Daniel Ruberman
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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