paper

Gluck twists on concordant or homotopic spheres

arXiv:2206.14113 · doi:10.4310/MRL.2023.v30.n6.a6

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 $I ¨ _{1} (M)$ 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.

14 pages, 1 figure, 1 table; v2: some typos corrected; v3: minor changes following a referee report, to appear in Mathematical Research Letters