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Locally equivalent Floer complexes and unoriented link cobordisms

Alberto Cavallo

Abstract

We show that the local equivalence class of the collapsed link Floer complex $cCFL^\infty(L)$, together with many $Υ$-type invariants extracted from this group, is a concordance invariant of links. In particular, we define a version of the invariants $Υ_L(t)$ and $ν^+(L)$ when $L$ is a link and we prove that they give a lower bound for the slice genus $g_4(L)$. Furthermore, in the last section of the paper we study the homology group $HFL'(L)$ and its behaviour under unoriented cobordisms. We obtain that a normalized version of the $\upsilon$-set, introduced by Ozsváth, Stipsicz and Szabó, produces a lower bound for the 4-dimensional smooth crosscap number $γ_4(L)$.

Locally equivalent Floer complexes and unoriented link cobordisms

Abstract

We show that the local equivalence class of the collapsed link Floer complex , together with many -type invariants extracted from this group, is a concordance invariant of links. In particular, we define a version of the invariants and when is a link and we prove that they give a lower bound for the slice genus . Furthermore, in the last section of the paper we study the homology group and its behaviour under unoriented cobordisms. We obtain that a normalized version of the -set, introduced by Ozsváth, Stipsicz and Szabó, produces a lower bound for the 4-dimensional smooth crosscap number .

Paper Structure

This paper contains 26 sections, 63 theorems, 186 equations, 31 figures.

Table of Contents

  1. Introduction
  2. Link Floer homology
  3. Chain complex and homology
  4. The Alexander and the F filtrations
  5. Duality and mirror images
  6. Local and stable equivalences of knot Floer chain complexes
  7. Concordance
  8. Canonical form of oriented link cobordisms
  9. Overview on grid diagrams
  10. Extended birth moves
  11. Extended death moves and invariance
  12. Upsilon-type invariants
  13. Definition of Upsilon*S(L) and the Upsilon-function for links
  14. Symmetries
  15. Connected sums and disjoint unions
  16. ...and 11 more sections

Key Result

Theorem 1.1

Consider two knots $K_1$ and $K_2$ in $S^3$. Then $CFK^\infty(K_1)$ is locally equivalent to $CFK^\infty(K_2)$ if and only if such two chain complexes are stably equivalent.

Figures (31)

  • Figure 1: Maslov gradings and algebraic filtration for $2$- (left) and $3$-component links (right). The algebraic level $j$ is on the $x$-axis and the Maslov grading on the $y$-axis.
  • Figure 2: The south-west region $W_{t,s}$ is the subset $\{(j,A)\:|\:j\leqslant t\text{ or }A\leqslant s\}$ of $\mathbb R^2$.
  • Figure 3: The dotted boundary in the picture on the right is not part of $\iota S$.
  • Figure 4: The complex $cCFL^\infty(T_{2,3})$ is on the left and $cCFL^\infty(T_{2,3}^*)$ on the right, both chain complexes are pictured ignoring the $U$-action. Black, white and gray dots represent Maslov gradings $1,0$ and $-1$ respectively.
  • Figure 5: Canonical form of oriented cobordisms between two links.
  • ...and 26 more figures

Theorems & Definitions (118)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Proposition 1.4
  • Theorem 1.5
  • Proposition 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Corollary 1.9
  • Theorem 1.10
  • ...and 108 more