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

$\def\spinc{\operatorname{spin}^\mathrm{c}}$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.

Full Text (PDF format)

Received 13 January 2022

Received revised 23 October 2022

Accepted 2 February 2023

Published 30 January 2024