The Simplicity Conjecture

The Simplicity Conjecture

Dan Cristofaro-Gardiner, University of California, Santa Cruz

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In the 60s and 70s, there was a flurry of activity concerning the question of whether or not various subgroups of homeomorphism groups of manifolds are simple, with beautiful contributions by Fathi, Kirby, Mather, Thurston, and many others. A funnily stubborn case that remained open was the case of area-preserving homeomorphisms of surfaces. For example, for balls of dimension at least 3, the relevant group was shown to be simple by work of Fathi in 1980; but, the answer in the two-dimensional case, asked in the 70s, was not known.

I will explain recent joint work proving that the group of compactly supported area preserving homeomorphisms of the two-disc is in fact not a simple group, which answers the ”Simplicity Conjecture” in the affirmative.

Our proof uses a new tool for studying area-preserving surface homeomorphisms, called periodic Floer homology (PFH) spectral invariants; these recover the classical Calabi invariant of monotone twists.