Duals of non-zero square

Abstract

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 both Gabai's and Scheniederman-Teichner's version 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.