Gluck twists on concordant or homotopic spheres

TL;DR

This paper investigates how Gluck twists along concordant or homotopic spheres in 4-manifolds affect the topology and smooth structure of the resulting manifolds, establishing conditions for homeomorphism and simple homotopy equivalence.

Contribution

It provides new results linking the concordance and homotopy of spheres to the topological and smooth classification of manifolds after Gluck twists, including necessary examples.

Findings

Concordant spheres yield s-cobordant manifolds after Gluck twists.

Homotopic spheres lead to simple homotopy equivalent manifolds.

Additional assumptions are needed for homeomorphism, demonstrated by counterexamples.

Abstract

Let $M$ be a compact $4$-manifold and let $S$ and $T$ be embedded $2$-spheres in $M$, both with trivial normal bundle. We write $M_S$ and $M_T$ for the $4$-manifolds obtained by the Gluck twist operation on $M$ along $S$ and $T$ respectively. We show that if $S$ and $T$ are concordant, then $M_S$ and $M_T$ are $s$-cobordant, and so if $\pi_1(M)$ is good, then $M_S$ and $M_T$ are homeomorphic. Similarly, if $S$ and $T$ are homotopic then we show that $M_S$ and $M_T$ are simple homotopy equivalent. Under some further assumptions, we deduce that $M_S$ and $M_T$ are homeomorphic. We show that additional assumptions are necessary by giving an example where $S$ and $T$ are homotopic but $M_S$ and $M_T$ are not homeomorphic. We also give an example where $S$ and $T$ are homotopic and $M_S$ and $M_T$ are homeomorphic but not diffeomorphic.