Gluck twists on concordant or homotopic spheres

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 th $M_S$ 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.

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D.K. was supported by the Deutsche Forschungsgemeinschaft under Germany’s Excellence Strategy - GZ 2047/1, Projekt-ID 390685813.

M.P. was partially supported by EPSRC New Investigator grant EP/T028335/1 and EPSRC New Horizons grant EP/V04821X/1.

Received 9 July 2022

Received revised 26 January 2023

Accepted 29 March 2023

Published 17 July 2024