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Instanton Floer homology, sutures, and Euler characteristics

Zhenkun Li, Fan Ye

Abstract

This is a companion paper to an earlier work of the authors. In this paper, we provide an axiomatic definition of Floer homology for balanced sutured manifolds and prove that the graded Euler characteristic $χ_{\rm gr}$ of this homology is fully determined by the axioms we proposed. As a result, we conclude that $χ_{\rm gr}(SHI(M,γ))=χ_{\rm gr}(SFH(M,γ))$ for any balanced sutured manifold $(M,γ)$. In particular, for any link $L$ in $S^3$, the Euler characteristic $χ_{\rm gr}(KHI(S^3,L))$ recovers the multi-variable Alexander polynomial of $L$, which generalizes the knot case. Combined with the authors' earlier work, we provide more examples of $(1,1)$-knots in lens spaces whose $KHI$ and $\widehat{HFK}$ have the same dimension. Moreover, for a rationally null-homologous knot in a closed oriented 3-manifold $Y$, we construct canonical $\mathbb{Z}_2$-gradings on $KHI(Y,K)$, the decomposition of $I^\sharp(Y)$ discussed in the previous paper, and the minus version of instanton knot homology $\underline{\rm KHI}^-(Y,K)$ introduced by the first author.

Instanton Floer homology, sutures, and Euler characteristics

Abstract

This is a companion paper to an earlier work of the authors. In this paper, we provide an axiomatic definition of Floer homology for balanced sutured manifolds and prove that the graded Euler characteristic of this homology is fully determined by the axioms we proposed. As a result, we conclude that for any balanced sutured manifold . In particular, for any link in , the Euler characteristic recovers the multi-variable Alexander polynomial of , which generalizes the knot case. Combined with the authors' earlier work, we provide more examples of -knots in lens spaces whose and have the same dimension. Moreover, for a rationally null-homologous knot in a closed oriented 3-manifold , we construct canonical -gradings on , the decomposition of discussed in the previous paper, and the minus version of instanton knot homology introduced by the first author.

Paper Structure

This paper contains 18 sections, 70 theorems, 256 equations, 17 figures.

Table of Contents

  1. Introduction
  2. Axioms and formal properties for sutured homology
  3. Axioms of a Floer-type theory for closed 3-manifolds
  4. Formal sutured homology of balanced sutured manifolds
  5. Gradings on formal sutured homology
  6. Heegaard Floer homology and the graph TQFT
  7. Heegaard Floer homology for multi-pointed 3-manifolds
  8. Cobordism maps for restricted graph cobordisms
  9. Floer's excision theorem
  10. Sutured Heegaard Floer homology
  11. The graded Euler characteristic of formal sutured homology
  12. Balanced sutured handlebodies
  13. Gradings about contact 2-handle attachments
  14. General balanced sutured manifolds
  15. The canonical mod 2 grading
  16. ...and 3 more sections

Key Result

Theorem 1.2

Suppose $(M,\gamma)$ is a balanced sutured manifold and $S_1,\dots,S_n$ are properly embedded admissible surfaces (c.f. Definition defn_2: admissible surfaces) generating $H_2(M,\partial M)/{\rm Tors}$. Then there exist $\mathbb{Z}^n$-gradings on $SHI(M,\gamma)$ and $SFH(M,\gamma)$ induced by these and a similar result holds for $SFH(M,\gamma)$. Moreover, there exist relative $\mathbb{Z}_2$-gradi

Figures (17)

  • Figure 1: The bypass triangle.
  • Figure 2: The disk $D$, the arcs $\beta_k,\beta_l,\beta_k',\beta_l'$, and the surface $U$.
  • Figure 3: The positive and negative stabilizations of $S$.
  • Figure 4: Free-stabilization in a small disk $D$.
  • Figure 5: Ribbon graph cobordisms related to free-stabilization maps.
  • ...and 12 more figures

Theorems & Definitions (143)

  • Conjecture 1.1: kronheimer2010knots
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Theorem 1.6
  • Remark 1.7
  • Remark 1.8
  • Remark 1.9
  • Theorem 1.10: LY2020
  • ...and 133 more