Some refined higher type adjunction inequalities on 4-manifolds

TL;DR

This paper refines higher type adjunction inequalities in 4-manifold topology, providing sharper bounds for embedded surfaces using Seiberg-Witten invariants, which enhances understanding of the topology of 4-manifolds.

Contribution

It improves existing adjunction inequalities for 4-manifolds with nonzero Seiberg-Witten invariants under specific conditions on embedded surfaces.

Findings

Sharper bounds on embedded surfaces in 4-manifolds.

Enhanced understanding of Seiberg-Witten invariants in topology.

Refined inequalities applicable to a broader class of surfaces.

Abstract

We further sharpen higher type adjunction inequalities of P. Ozsv\'ath and Z. Szab\'o on a 4-manifold $M$ with a nonzero Seiberg-Witten invariant for a Spin$^c$ structure $\frak{s}$, when an embedded surface $\Sigma\subset M$ satisfies $[\Sigma]\cdot [\Sigma]\geq 0$ and $$|\langle [\Sigma],c_1(\frak{s})\rangle|+[\Sigma]\cdot [\Sigma]\geq 2b_1(M).$$