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

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.