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Filtered instanton Floer homology and the homology cobordism group

Yuta Nozaki, Kouki Sato, Masaki Taniguchi

Abstract

For any $s \in [-\infty, 0] $ and oriented homology 3-sphere $Y$, we introduce a homology cobordism invariant $r_s(Y)\in (0,\infty]$. The values $\{r_s(Y)\}$ are included in the critical values of the $SU(2)$-Chern-Simons functional of $Y$, and we show a negative definite cobordism inequality and a connected sum formula for $r_s$. As applications, we obtain several new results on the homology cobordism group. First, we give infinitely many homology 3-spheres which cannot bound any definite 4-manifold. Next, we show that if the 1-surgery of $S^3$ along a knot has the Frøyshov invariant negative, then all positive $1/n$-surgeries along the knot are linearly independent in the homology cobordism group. In another direction, we use $\{r_s\}$ to define a filtration on the homology cobordism group which is parametrized by $[0,\infty]$. Moreover, we compute an approximate value of $r_s$ for the hyperbolic 3-manifold obtained by $1/2$-surgery along the mirror of the knot $5_2$.

Filtered instanton Floer homology and the homology cobordism group

Abstract

For any and oriented homology 3-sphere , we introduce a homology cobordism invariant . The values are included in the critical values of the -Chern-Simons functional of , and we show a negative definite cobordism inequality and a connected sum formula for . As applications, we obtain several new results on the homology cobordism group. First, we give infinitely many homology 3-spheres which cannot bound any definite 4-manifold. Next, we show that if the 1-surgery of along a knot has the Frøyshov invariant negative, then all positive -surgeries along the knot are linearly independent in the homology cobordism group. In another direction, we use to define a filtration on the homology cobordism group which is parametrized by . Moreover, we compute an approximate value of for the hyperbolic 3-manifold obtained by -surgery along the mirror of the knot .

Paper Structure

This paper contains 37 sections, 75 theorems, 224 equations, 14 figures, 2 tables.

Table of Contents

  1. Introduction
  2. The invariants $r_s$
  3. Topological applications
  4. Homology $3$-spheres with no definite bounding
  5. Linear independence of $1/n$-surgeries
  6. Linear independence of Whitehead doubles
  7. Additional structures on $\Theta^3_\mathbb Z$ and $\operatorname{Ker} h$
  8. Computations for a hyperbolic 3-manifold
  9. Review of filtered instanton Floer homology
  10. Preliminaries
  11. Chern-Simons functional
  12. Perturbations of $\mathit{cs}_Y$
  13. Gradient of $\mathit{cs}_{Y, \pi}$
  14. Classes of perturbations
  15. Instanton Floer homology
  16. ...and 22 more sections

Key Result

Theorem 1.1

The values $\{r_s(Y)\}_{s \in [-\infty, 0]}$ are homology cobordism invariants of $Y$. Moreover, the invariants $\{r_s\}_{s \in [-\infty, 0]}$ satisfy the following properties:

Figures (14)

  • Figure 1: A schematic picture of the filtration $\{\Theta^3_{\mathbb Z,r}\}$.
  • Figure 2: The knot $K_k$.
  • Figure 3: Kirby calculus for $S^3_{-1/k}(K_1) \cong S^3_{-1}(K_k)$.
  • Figure 4: The cobordism $W_n$.
  • Figure 5: The 4-manifold $X_n$.
  • ...and 9 more figures

Theorems & Definitions (144)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Corollary 1.7
  • Theorem 1.8
  • Corollary 1.9
  • Corollary 1.10
  • ...and 134 more