The non-simply connected Price twist for the 4-sphere

TL;DR

This paper investigates the properties and diffeomorphism types of a specific non-simply connected 4-manifold obtained via Price twist on P^2-knots in the 4-sphere, expanding understanding of 4-manifold topology.

Contribution

It characterizes the diffeomorphism types of the non-simply connected manifolds resulting from Price twists on Kinoshita-type P^2-knots in the 4-sphere.

Findings

Identifies conditions under which the Price twist yields non-simply connected manifolds.

Classifies the diffeomorphism types of these manifolds.

Provides new insights into the topology of 4-manifolds with non-trivial fundamental groups.

Abstract

A cutting and pasting operation on a $P^2$-knot $S$ in a $4$-manifold is called the Price twist. The Price twist for the $4$-sphere $S^4$ yields at most three $4$-manifolds up to diffeomorphism, namely, the $4$-sphere $S^4$, the other homotopy $4$-sphere $\Sigma_{S}(S^4)$ and a non-simply connected $4$-manifold $\tau_{S}(S^4)$. In this paper, we study some properties and diffeomorphism types of $\tau_{S}(S^4)$ for $P^2$-knots $S$ of Kinoshita type.