Caltech/USC/UCLA Joint Topology Seminar
I'll discuss recent work with Irving Dai and Abhishek Mallick in which we study involutions on homology spheres, up to a natural notion of cobordism. Using this notion, we define a 3-dimensional homology bordism group of diffeomorphisms which refines both the homology cobordism group and the bordism group of diffeomorphisms. The subgroup generated by involutions provides a new algebraic framework in which to study corks: contractible 4-manifolds equipped with involutions on their boundaries which do not extend smoothly to their interiors. Using Heegaard Floer homology, we construct invariants of manifolds with involutions in much the same spirit as involutive Floer homology. We use these invariants to study corks and demonstrate that, very often, the involutions on their boundary do not extend over any contractible 4-manifold. I'll discuss a number of such examples.