Trisections of surface complements and the Price twist

TL;DR

This paper develops methods to describe Price twists and surface complements in 4-manifolds using trisections, facilitating the study of exotic smooth structures and homotopy spheres.

Contribution

It introduces a way to produce trisection descriptions of Price twists and general surface complements in 4-manifolds, advancing the understanding of 4-manifold topology.

Findings

Constructed trisection descriptions for Price twists on surfaces in 4-manifolds.

Provided a method to produce relative trisections of surface complements.

Enabled analysis of exotic smooth structures via trisection techniques.

Abstract

Given an $S\cong \mathbb{R}P^2$ smoothly embedded in a 4-manifold $X^4$ with Euler number 2 or -2, the Price twist is a surgery operation on $\nu(S)$ yielding (up to) three different 4-manifolds: $X^4,\tau_S(X^4),\Sigma_S(X^4)$. This is of particular interest when $X^4=S^4$, as then $\Sigma_S(X^4)$ is a homotopy 4-sphere which is not obviously diffeomorphic to $S^4$. In this paper, we show how to produce a trisection description of each Price twist on $S\subset X^4$ by producing a relative trisection of $X^4\setminus\nu(S)$. Moreover, we show how to produce a trisection description of general surface complements in 4-manifolds.