Duals of non-zero square

TL;DR

This paper constructs specific 4-manifolds with pairs of spheres having a common dual of a given square, illustrating the necessity of the square zero condition in certain 4D Light Bulb Theorem versions and revealing invariants that vanish.

Contribution

It provides explicit examples of 4-manifolds with pairs of spheres that challenge existing theorems, highlighting the importance of the dual's square condition and connecting to the Mazur cork.

Findings

Examples show the square zero assumption is necessary in the 4D Light Bulb Theorem.

Both Freedman-Quinn and Kervaire-Milnor invariants vanish for these sphere pairs.

Application of results related to the Mazur cork in the proof.

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 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.