Non-isotopic surfaces in $T^4\#(S^2\times S^2)$: an example

TL;DR

This paper constructs infinitely many embedded tori in a specific 4-manifold that are homotopic and diffeomorphic but not isotopic, even after stabilization, distinguished by their handle complement homotopy classes.

Contribution

It provides explicit examples of non-isotopic embedded tori with a common geometric dual in a complex 4-manifold, using the Norman trick and handle analysis.

Findings

Existence of infinitely many non-isotopic tori with the same homotopy class.

These tori are distinguished by homotopy classes of 2-handles in their complements.

Non-isotopic tori persist even after arbitrary external stabilizations.

Abstract

We prove that there exist infinitely many embedded tori with a common geometric dual in $T^4\#(S^2\times S^2)$ that are homotopic, diffeomorphic, but not isotopic to each other, even after arbitrary many external stabilizations. These surfaces are obtained by applying the Norman trick to a fixed immersed surface, using non-homotopic tubing arcs. The isotopy classes of these surfaces are distinguished by homotopy classes of the 2-handles (relative to the boundary) in the complement of the image of the $0$- and $1$-handles.