Gluck twists on concordant or homotopic spheres

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 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.

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