Bordism und projective space bundles

TL;DR

This paper proves that total spaces of complex projective plane bundles generate the oriented cobordism ring, providing a new proof that certain simply connected manifolds admit positive scalar curvature metrics.

Contribution

It establishes that $CP^2$-bundles generate the oriented cobordism ring and offers an alternative proof of positive scalar curvature existence for specific manifolds.

Findings

$CP^2$-bundles generate the oriented cobordism ring

New proof of positive scalar curvature existence for simply connected non-spin manifolds

Connections between cobordism theory and scalar curvature results

Abstract

We prove that total spaces of $CP^2$-bundles generate the oriented cobordism ring $\Omega_*$. Combined with the surgery lemma this yields a somewhat different proof of Gromov's and Lawson's theorem that all simply connected non-spin manifolds of dimension $\geq$ 5 carry a metric of positive scalar curvature.