Gluck twists on concordant or homotopic spheres
Daniel Kasprowski, Mark Powell, Arunima Ray · Mathematical Research Letters · 2023
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 π 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.