https://rochester.zoom.us/j/6775967436

This talk will describe a diffeomorphism of “the barbell manifold” and what it tells us about smooth isotopy of 3-manifolds in some small 4-manifolds. Specifically, the “barbell” is the (4,2)-handlebody of genus 2, i.e. the boundary connect-sum of two copies of S^2 x D^2. We show that the mapping class group of the barbell manifold, i.e. \pi_0 Diff(Barbell), where the diffeomorphisms fix the boundary pointwise, is infinite cyclic – after perhaps modding out by the mapping class group of D^4. We then consider embedding the barbell into various small 4-manifolds, and the question of whether or not the natural extension of the barbell diffeomorphism is isotopically trivial in these 4-manifolds. From this we can conclude that the mapping class groups of both S^1 x D^3 and S^1 x S^3 are not finitely generated. For S^1 x D^3 the idea of the proof is to show these diffeomorphisms act non-trivially on the isotopy classes of reducing 3-balls, i.e. show f({1}xD^3) is not isotopic to {1}xD^3. To do this, we imagine D^3 as a 2-parameter family of intervals, thus f({1}xD^3) can be viewed as producing an element of the 2nd homotopy group of the space of smooth embeddings of an interval in S^1 x D^3. The core of the proof involves developing an invariant that can detect the low-dimensional homotopy groups of embedding spaces. These invariants can be thought of as Vassiliev invariants.