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.

Counterexamples in 4-manifold topology

“…Whether this is possible in the orientable setting remains open. Other examples of the operation changing the smooth structure on nonorientable 4-manifolds are given in [135], [71,Proposition 1.6]. See also [10] for a condition that implies the Gluck twist operation does not change the diffeomorphism type.…”