Bordered Floer homology, handlebody detection, and compressing diffeomorphisms

TL;DR

This work shows that bordered Floer invariants, including twisted and bordered-sutured versions, detect when a 3-manifold is a handlebody and determine whether a surface diffeomorphism extends over a given handlebody or compression body. By combining bordered-sutured Floer theory with train-track/arc-diagram technology and adapting Casson–Long-style arguments, the authors produce an algorithmic framework to decide extension questions, with explicit bounds on the complexity of the relevant bimodules and Heegaard diagrams. Central to the approach are the notions of the support of twisted invariants and its relationship to the presence of homologically essential $2$-spheres, as well as a sequence of train-track-driven arcslides that encode mapping classes. The results establish computability of the detection problems and provide concrete, quantitative bounds linking geometric data (genus, boundary components, and train-track features) to the algebraic complexity of bordered-sutured invariants, enabling practical algorithms for extension questions in 3-manifold topology. The methods unify train-track dynamics, Floer theoretic obstructions, and bordered-sutured calculus to produce a robust toolkit for handlebody and compression-body detectability with potential broader applications in 3-manifold topology and mapping class group algorithms.

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.

Figures (28)

• Figure 1: Switches. Left: a general switch. Right: a trivalent switch, with incident half-branches $b_1$, $b_2$, and $b_3$. The half-branches $b_1$ and $b_2$ are small and $b_3$ is large. The switch condition for a measure $\mu$ is that $\mu(b_3)=\mu(b_1)+\mu(b_2)$.
• Figure 2: A split. There are three cases, depending on the relative weights of the branches.
• Figure 3: A shift. The weight of the inner edge is the sum of the weights of the two edges to its right; the other weights are unchanged by the move.
• Figure 4: The image of a 4-gon $B$ under $\psi$. Thick lines indicate the train track $\tau$. Thin lines indicate the boundary of a fibered neighborhood of $\tau$. The shaded region is $B$ (left) and the image of $B$ (right). The image of $B$ and the train paths $\gamma_i$ extend beyond the picture. (In the drawing, near the singular point $\psi$ sends $B$ by a $135^\circ$ twist.)
• Figure 5: Finding a loop. Left: if $\gamma$ goes over a non-infinitesimal branch twice in the same direction then it contains a loop. Center: if $\gamma$ goes over the same diagonal twice then it goes over the same non-diagonal edge twice. Right: if $\gamma$ goes over two diagonals incident to the same switch and goes over the adjacent branch twice in opposite directions, then $\gamma$ can be short-circuited by a different infinitesimal diagonal to give a loop. In the center and right pictures, dotted edges indicate diagonals and solid edges the original train track $\tau$; the original train path under consideration is thick. In the left picture, edges could either be diagonals or in $\tau$.

Theorems & Definitions (94)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Theorem 1.5
• Corollary 1.6
• Lemma 2.1
• proof
• Theorem 2.2
• Lemma 2.3
• proof