Pure and Applied Mathematics Quarterly

Volume 19 (2023)

Number 6

Special Issue in honor of Professor Blaine Lawson’s 80th birthday

Guest Editors: Shiu-Yuen Cheng, Paulo Lima-Filho, and Stephen Shing-Toung Yau

Any oriented non-closed connected $4$-manifold can be spread holomorphically over the complex projective plane minus a point

Pages: 2915 – 2918

DOI: https://dx.doi.org/10.4310/PAMQ.2023.v19.n6.a11

Author

Dennis Sullivan (City University of New York, N.Y., U.S.A.; and Stony Brook University, Stony Brook, New York, U.S.A.)

Abstract

$\def\spinc{\operatorname{spin}^\mathrm{c}}$We give a 1965 era proof of the title assuming $M$ is $\spinc$. The fact that any oriented four manifold is $\spinc$ is a challenging result from 1995 whose interesting argument by Teichner–Vogt is analyzed and used in the appendix to show an analogous integral lift result about the top Wu class in $\dim 4k$. This will be used in future work to study related complex structures on higher dimensional open manifolds.

Keywords

$\operatorname{spin}^{\mathrm{c}$, complex structures, $4$-manifolds

2010 Mathematics Subject Classification

Primary 57R15. Secondary 57-xx.

Received 13 January 2022

Received revised 23 October 2022

Accepted 2 February 2023

Published 30 January 2024