# Isotopy vs. homotopy for disks with a common dual

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Hannah Schwartz, Princeton University

Recent work of both Gabai and Schneiderman-Teichner on the smooth isotopy of homotopic surfaces with a common dual has reinvigorated the study of concordance invariants defined by Freedman and Quinn in the 90's, along with homotopy theoretic invariants of Dax from the 70's. We will show that in any smooth $4$-manifold with boundary, pairs of properly embedded disks related by homotopy supported away from a common dual are in fact isotopic, if and only if the `Dax invariant' of the pair vanishes.