Minimal genus for 4-manifolds with $b^+=1$

TL;DR

This paper establishes a new adjunction inequality for 4-manifolds with $b^+=1$, generalizing previous results and providing powerful constraints based on cohomology and topological invariants.

Contribution

It introduces a generalized adjunction inequality for 4-manifolds with $b^+=1$ that depends solely on the cohomology algebra, extending prior inequalities.

Findings

The inequality applies to manifolds with $b^+=1$ and depends only on cohomology.

It generalizes Strle's inequality for cases with $b_1=0$.

The inequality is especially effective when $2\tilde \chi+3\sigma \geq 0$.

Abstract

We derive an adjunction inequality for any smooth, closed, connected, oriented 4-manifold $X$ with $b^+=1$. This inequality depends only on the cohomology algebra and generalizes the inequality of Strle in the case of $b_1=0$. We demonstrate that the inequality is especially powerful when $2\tilde \chi+3\sigma\geq 0$, where $\tilde \chi$ is the modified Euler number taking account of the cup product on $H^1$.