Georgia Institute of Technology College of Sciences

This is joint work with Jen Hom and Tye Lidman. Attaching a Casson handle to a slice disk complement gives a smooth manifold homeomorphic to R^4. In the 90's De Michelis and Freedman asked how these choices affect the smooth type of the resulting manifold. This problem has seen some progress since then but is still not well understood. We show that if two slice knots have sufficiently different knot Floer homology, then the resulting exotic R^4's made with the simplest Casson handle are distinct. This gives a countably infinite family of exotic R^4's made with different slice disk complements. We then produce exotic R^4's with various phenomena, and re-prove a theorem of Bizaca-Etnyre on smoothings of product manifolds Y x R. Our main tool is Gadgil's end Floer homology, which we show how to compute effectively by analyzing a certain cobordism map. Time permitting, I'll discuss an upcoming result on exotic planes in R^4 and branched covers, and plans to study more noncompact exotic phenomena.