Bordered Floer homology, handlebody detection, and compressing diffeomorphisms

TL;DR

This paper demonstrates that bordered Floer homology can detect handlebodies and whether a mapping class extends over them, providing an algorithm for such detection using advanced topological tools.

Contribution

It introduces a novel application of bordered Floer homology to handlebody detection and extension problems, combining Casson-Long ideas with train track theory.

Findings

Bordered Floer homology detects handlebodies up to connected sums with L-spaces.

An algorithm is developed to determine if a mapping class extends over a compression body.

The approach integrates topological and algebraic methods for handlebody detection.

Abstract

We show that, up to connected sums with integer homology $L$-spaces, bordered Floer homology detects handlebodies, as well as whether a mapping class extends over a given handlebody or compression body. Using this, we combine ideas of Casson-Long with the theory of train tracks to give an algorithm using bordered Floer homology to detect whether a mapping class extends over any compression body.