Light bulbs in 4-manifolds

Maggie Miller (Princeton University)

Monday 13th January, 2020 16:00-17:00 Maths 311B

Abstract

 In 2017, Gabai proved the light bulb theorem, showing that
if $R$ and $R'$ are 2-spheres homotopically embedded in a 4-manifold
with a common dual, then with some condition on 2-torsion in
$\pi_1(X)$ one can conclude that $R$ and $R'$ are smoothly isotopic.
Schwartz later showed that this 2-torsion condition is necessary, and
Schneiderman and Teichner then obstructed the isotopy whenever this
condition fails. I weakened the hypothesis on $R$ and $R'$ by allowing
$R'$ not to have a dual and showed that the spheres are still smoothly
concordant.

I will talk about these various definitions and theorems as well as
current joint work with Michael Klug generalizing the result on
concordance to the situation where $R$ has an immersed dual (and $R'$
may have none), which is a common condition in 4-dimensional topology.

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