Asymptotic behavior of invariants of homology spheres

As shown by Morita, every integral homology 3-sphere Y has a decomposition into two simple pieces (called a Heegaard splitting) glued along a surface diffeomorphism which acts trivially on the homology of the surface. These diffeomorphisms form the Torelli subgroup of the mapping class group of the surface, and the Torelli group is finitely generated for surfaces of genus 3 or higher. Though the group of integral homology spheres is infinitely generated, by fixing the genus of a Heegaard splitting we can use finite generation in the surface setting to better understand how invariants like the Rokhlin and Casson invariant change. This perspective has led to important results in the study of the Torelli group and the Casson invariant in work of Birman-Craggs-Johnson, Morita, and Broaddus-Farb-Putman. In work in progress with Santana Afton and Tye Lidman, we show that the d-invariant from Heegaard Floer homology of an integral homology sphere is bounded above by a linear function of the word length of a corresponding gluing map in the Torelli group. Moreover, we show the d-invariant is bounded for homology spheres corresponding to various large families of mapping classes. If time permits, we will discuss the case of rational homology spheres.