Concordance of light bulbs

Maggie Miller (Princeton University, Mathematics)

Monday, December 2, 2019 -
3:15pm to 4:15pm
Location:
119 Physics

In 2017, Gabai proved the light bulb theorem: if R and R' are 2-spheres in a 4-manifold X which are homotopic and have a common dual (i.e. R and R' are “light bulbs”), then (modulo a statement about 2-torsion in \pi_1(X)), they are smoothly isotopic. (This setting is motivated by handle cancellation of smooth cobordisms of 4-manifolds.) Schwartz later showed that this 2-torsion hypothesis is necessary, and Schneiderman and Teichner then gave an obstruction in the form of an invariant fq associated to a pair of homotopic spheres that vanishes on light bulbs if and only if they are isotopic. I consider 2-spheres R and R’ in a 4-manifold $X$ which are homotopic, but when only $R$ has a dual (i.e. R and R’ are “algebraic light bulbs”). In this setting, I prove that R and R’ are smoothly concordant with the same 2-torsion hypothesis as Gabai -- in general, they need not be isotopic. (This setting is motivated by handle cancellation of topological cobordisms of 4-manifolds.) I will talk about this work as well as current joint work with Michael Klug redefining fq to prove that this invariant vanishes on algebraic light bulbs if and only if they are concordant.

Last updated: 2020/04/04 - 2:36pm