Degree-1 maps and rank inequalities in Heegaard Floer homology

TL;DR

We study degree-1 maps from rational homology solid tori to mapping-torus targets $E_n$, focusing on the case $n=2$ where $E_2$ is the twisted $I$-bundle over the Klein bottle, to derive rank inequalities for $\widehat{HF}$. The approach extends the immersed-curve interpretation of bordered Floer homology by analyzing rational-homology solid tori with rational longitudes of order $2$ and comparing their immersed curves via lifted geodesics on a marked torus, turning rank questions into Lagrangian intersection computations. We prove that if $M$ is obtained from a rational homology sphere $N$ by a $2$-pinch, then $\operatorname{rank}(\widehat{HF}(M))\le \operatorname{rank}(\widehat{HF}(N))$, and we formulate a precise generalization to $n$-pinches under suitable spin$^c$-restrictions. These results bolster Kronheimer–Mrowka's conjectured relation between Heegaard Floer and instanton Floer homologies and illustrate how bordered Floer tools translate degree-1 maps into rank inequalities.

Abstract

Ghosh-Sivek-Zentner constructed degree-1 maps from certain rational homology solid tori to the twisted $I$-bundle over the Klein bottle. We show that these maps yield rank inequalities for Heegaard Floer homology. To do so, we use Hanselman-Rasmussen-Watson's immersed curve interpretation of bordered Floer homology, extending their proof of a similar rank inequality corresponding to degree-1 maps to the solid torus. Our result provides further evidence for Kronheimer-Mowka's conjectured relationship between Heegaard Floer homology and instanton Floer homology.

Figures (7)

• Figure 1: The monodromy $\phi_n: P_n \to P_n$.
• Figure 2: The component for $(d_0^*)^{4}$. More generally, the component for $E_n$ is a loop of $n$ arrows each decorated with $\rho_{12}$.
• Figure 3: The component for $(a_2^*)(b_{-2}^*)$ in $E_4$. The corresponding component for $E_{2n}$ can be obtained by adding an extra $n-2$$\rho_{12}$ arrows in the middle of the upper and lower rows of the diagram.
• Figure 4: A component of the immersed curve invairant of $E_4$ corresponding to a self-conjugate $\text{spin}^c$-structure.
• Figure 5: A component of the immersed curve of $E_4$ corresponding to a self-conjugate $\text{spin}^c$-structure. To obtain the corresponding component of the immersed curve for $E_{2n}$, replace the two basepoints above (below) the horizontal row of black dots with $n$ evenly placed black dots.

Theorems & Definitions (18)

• Theorem 1.1: GSZdegree-1
• Theorem 1.2
• Conjecture 1.3
• Example 2.1
• Lemma 2.2: hanselman2016bordered
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3