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Seminar

FHT Program Seminar: Exotically Embedded Submanifolds in 4-Manifolds and Stabilizations October 06, 2022 (01:15 PM PDT - 02:00 PM PDT)
Parent Program:
Location: MSRI: Simons Auditorium, Online/Virtual
Speaker(s) Anubhav Mukherjee (Georgia Institute of Technology)
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FHT Program Seminar: Exotically Embedded Submanifolds In 4-Manifolds And Stabilizations

Abstract/Media

To participate in this seminar, please register HERE.

An important principle in 4-dimesional topology, as discovered by Wall in the 1960s, states that all exotic phenomena are eliminated by sufficiently many stabilizations (i.e., taking connected sum with S2 xS2's). Since then, it has been a fundamental problem to search for exotic phenomena that survives one stabilization. In this talk, we will establish the first pair of orientable exotic surfaces (in a puctured K3) which are not smoothly isotopic even after one stabilization. A key ingredient in our argument is a vanishing theorem for the family Bauer-Furuta invariant, proved using equivariant stable homotopy theory. This theorem applies to a large family of spin 4-manifolds and has some interesting applications in Smale's conjecture (about exotic diffeomorphisms on S^4). In particular, it implies that the S^1-equivariant or non-equivariant family Bauer-Furuta invariant do not detect an exotic diffeomorphism on S^4 and it suggests that the Pin(2)-symmetry could be a game changer. This is a joint work with Jianfeng Lin. Also if time permits I will discuss exotically embedded 3-manifolds in 4-manifolds which is an upcoming work joint with Hokuto Konno and Masaki Taniguchi.

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FHT Program Seminar: Exotically Embedded Submanifolds In 4-Manifolds And Stabilizations