Duals of non-zero square

In this short note, for each non-zero integer $n$, we construct a $4$-manifold containing a smoothly concordant pair of spheres with a common dual of square $n$ but no automorphism carrying one sphere to the other. Our examples, besides showing that the square zero assumption on the dual is necessary in Gabai’s and Schneiderman–Teichner’s versions of the 4D Light Bulb Theorem, have the interesting feature that both the Freedman–Quinn and Kervaire–Milnor invariant of the pair of spheres vanishes. The proof gives a surprising application of results due to Akbulut–Matveyev and Auckly–Kim-Melvin–Ruberman pertaining to the well-known Mazur cork.

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Received 27 December 2020

Accepted 1 June 2021

Published 6 September 2022