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On homology spheres of the trivial local equivalence class

Jaewon Lee, Oğuz Şavk

Abstract

In the homology cobordism group $Θ_\mathbb{Z}^3$, it is not known if there are non-trivial linear dependences between Seifert fibered spheres. Based on involutive Heegaard Floer theory, Hendricks, Manolescu, and Zemke introduced the local equivalence group $\mathfrak{I}$ along with the homomorphism $h:Θ_\mathbb{Z}^3 \rightarrow \mathfrak{I}$. Using the work of Dai and Stoffregen, one can find non-trivial linear dependences between the images of Seifert fibered spheres under $h$. Therefore, it is interesting to ask if such dependences in $\mathfrak{I}$ originate from $Θ_\mathbb{Z}^3$. In this paper, by employing the $r_s$-invariants from the filtered instanton Floer homology developed by Nozaki, Sato, and Taniguchi, we provide certain conditions to guarantee that such relations are not realized even in the rational homology cobordism group. We also discuss the local equivalence class of the $\operatorname{Pin(2)}$-equivariant Seiberg--Witten Floer stable homotopy type.

On homology spheres of the trivial local equivalence class

Abstract

In the homology cobordism group , it is not known if there are non-trivial linear dependences between Seifert fibered spheres. Based on involutive Heegaard Floer theory, Hendricks, Manolescu, and Zemke introduced the local equivalence group along with the homomorphism . Using the work of Dai and Stoffregen, one can find non-trivial linear dependences between the images of Seifert fibered spheres under . Therefore, it is interesting to ask if such dependences in originate from . In this paper, by employing the -invariants from the filtered instanton Floer homology developed by Nozaki, Sato, and Taniguchi, we provide certain conditions to guarantee that such relations are not realized even in the rational homology cobordism group. We also discuss the local equivalence class of the -equivariant Seiberg--Witten Floer stable homotopy type.

Paper Structure

This paper contains 7 sections, 17 theorems, 44 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries on Floer Theories
  3. Involutive Heegaard Floer homology and local equivalence group
  4. Monotone graded subroots for Seifert fibered spheres
  5. Local equivalence in Seiberg--Witten Floer theory
  6. Filtered instanton Floer homology and rs-invariants
  7. Proof of Theorem \ref{['thm-Z']} and Examples

Key Result

Theorem 1.2

Let $Y$ be a Seifert fibered sphere and let $Z_Y$ be the associated homology sphere of the same image in $\mathfrak{I}$. If $R(Y) > 0$ and $\mathcal{E} (Y)$ is distinct from $30$ and $(2m+1)(4m+1)(4m+3)$, then $Y\# -Z_Y$ is non-trivial in $\ker{h}$. Moreover, given such a family of homology spheres

Figures (4)

  • Figure 1: The graded root $R$ with the involution $J$ for the Brieskorn sphere $\Sigma(3,4,13)$. The leaves and angles of $R$ are labeled by $v_1, \ldots v_6$ and $\alpha_1, \ldots \alpha_5$, respectively.
  • Figure 2: The standard complex $C_*(R)$ with the involution $J$ that captures the lattice homology $\mathbb{H}^- (\Sigma(3,4,13))$. Here, the solid (resp. the dashed) lines represent the action of $U$ (resp. $\partial$).
  • Figure 3: For the Brieskorn sphere $\Sigma(3,4,13)$, the monotone graded subroot $M = M(0,-2)$ with the involution $J$ is drawn in black, compare with Figure \ref{['fig:graded_root']}.
  • Figure 4: The monotone graded subroots of $B(n)$, $Y_1(n)$ and $Y_2(n)$ for $n \geq 1$.

Theorems & Definitions (25)

  • Theorem 1.2
  • Corollary 1.3
  • Theorem 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4: Theorems 4.5 and 6.1, DM19
  • Theorem 2.5: Section 8, DM19; Theorem 4.4, DS19
  • Theorem 2.6: Theorem 4.2, DS19
  • Theorem 2.7: Theorem 1.1, DS19
  • Theorem 2.8: Theorems 1.2 and 7.3, DSS23
  • ...and 15 more