# Comparing homological invariants for mapping classes of surfaces

## TL;DR

This paper compares two types of surface mapping class invariants, Hochschild homology from bordered Floer theory and fixed point Floer cohomology, proposing a conjectural isomorphism supported by genus two computations.

## Contribution

It introduces a comparison between Hochschild homology invariants and fixed point Floer cohomology for surface mapping classes, including explicit genus two calculations and a conjectural isomorphism.

## Key findings

- Computed Hochschild homology for genus two mapping class bimodules
- Observed similarities between Hochschild and fixed point Floer cohomology
- Proposed a conjectural isomorphism between the two invariants

## Abstract

We compare two different types of mapping class invariants: the Hochschild homology of an $A_\infty$ bimodule coming from bordered Heegaard Floer homology, and fixed point Floer cohomology. We first compute the bimodule invariants and their Hochschild homology in the genus two case. We then compare the resulting computations to fixed point Floer cohomology, and make a conjecture that the two invariants are isomorphic. We also discuss a construction of a map potentially giving the isomorphism. It comes as an open-closed map in the context of a surface being viewed as a $0$-dimensional Lefschetz fibration over the complex plane.