Ribbon Homology Cobordisms and Link Floer Homology

TL;DR

This work investigates how decorated link cobordisms in ribbon $\mathbb{Z}$-homology cobordisms act on link Floer homology. By combining results from Daemi–Lidman–Vela-Vick–Wong and Zemke, it shows that ribbon $\mathbb{Z}$-homology concordances induce split injections on $\mathcal{HFL}^-$ and develops a torsion-based framework with the torsion order $\mathrm{Ord}_V$ to bound the number of critical points via $2g(\Sigma)$. The paper derives inequalities relating torsion orders across ribbon cobordisms, yielding Seifert genus bounds and fusion-number obstructions for ribbon knots, and it clarifies how the $H_1$ action and graph TQFT organize the extended link cobordism maps. Together, these results connect decorated link cobordisms, the $H_1$-action, and the graph cobordism formalism in a way that yields concrete structural constraints for ribbon cobordisms between links. The framework has potential to constrain ribbon cobordism pathways and to inform invariants in 3- and 4-dimensional topology via link Floer theory.

Abstract

We make use of link Floer homology to study cobordisms between links embedded in 4-dimensional ribbon homology cobordisms. Combining results of Daemi--Lidman--Vela-Vick--Wong and Zemke, we show that ribbon homology concordances induce split injections on $\mathcal{HFL}^-$. We also make use of the torsion submodule of $\text{HFL}^-$ to give restrictions on the number of critical points in ribbon homology concordances.

Figures (6)

• Figure 2.1: The graph $\Gamma$ realizing the action of a closed curve $\gamma$ in $Y$. The cyclic ordering is indicated by the dashed arrow.
• Figure 2.2: The decorated link cobordisms $(S^1\times D^3, M)$ (left) and $(D^2 \times S^2, M')$ (right).
• Figure 3.1: A schematic of the decorated surface in $S^2\times S^1\times \{0\}$ described in the text preceding Lemma \ref{['Lemma: H1 Action on S1x S2']}.
• Figure 3.2: A schematic of the decorated surface $\mathcal{F}_\gamma$. In general, $K\times \{0\}$ and $\gamma$ might be linked.
• Figure 3.3: When the link cobordism map on the right is followed by the action of $\gamma$, it becomes equivalent to the map on the left.

Theorems & Definitions (42)

• Definition 1.1
• Remark 1.2
• Definition 1.3
• Definition 1.4
• Theorem 1
• Corollary 1.5
• Definition 1.6
• Theorem 2
• Definition 2.1
• Definition 2.2