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Finite type invariants in low degrees and the Johnson filtration

Wolfgang Pitsch, Ricard Riba

TL;DR

This work analyzes how finite-type invariants of integral homology spheres interact with the Johnson filtration of the mapping class group. It shows that while certain degree-1 and degree-2 invariants behave as homomorphisms on early Johnson levels (with $\lambda$ and $d_2$ playing key roles), a specific linear combination of invariants, $d_2+3\lambda$, vanishes on the fifth Johnson level, obstructing the realization of some spheres (e.g., the Poincaré sphere) via level-5 gluing maps. The authors derive a surgery formula for Ohtsuki's $\lambda_2$ by expressing its associated 2-cocycle as a bilinear form pulled back through Johnson homomorphisms, and they explicitly determine the bilinear coefficients that govern $\lambda_2$ in terms of $\tau_2$ and related constructs. Collectively, these results refine the understanding of finite-type invariants in low degree and illuminate the limitations of low-level Johnson filtrations to capture all homology spheres.

Abstract

We study the behaviour of the Casson invariant $λ$, its square, and Othsuki's second invariant $λ_2$ as functions on the Johnson subgroup of the mapping class group. We show that since $λ$ and $d_2 = λ_2 - 18 λ^2$ are invariants that are morphisms on respectively the second and the third level of the Johnson filtration they never vanish on any level of this filtration. In contrast we prove that the invariant $λ_2-18λ^2 +3λ$ vanishes on the fifth level of the Johnson filtration, $\mathcal{M}_{g,1}(5)$, and as a consequence we prove that, for instance, the Poincaré homology sphere does not admit any Heegaard splitting with gluing map an element in $\mathcal{M}_{g,1}(5)$. Finally we determine a surgery formula for Othsuki's second invariant $λ_2$.

Finite type invariants in low degrees and the Johnson filtration

TL;DR

This work analyzes how finite-type invariants of integral homology spheres interact with the Johnson filtration of the mapping class group. It shows that while certain degree-1 and degree-2 invariants behave as homomorphisms on early Johnson levels (with and playing key roles), a specific linear combination of invariants, , vanishes on the fifth Johnson level, obstructing the realization of some spheres (e.g., the Poincaré sphere) via level-5 gluing maps. The authors derive a surgery formula for Ohtsuki's by expressing its associated 2-cocycle as a bilinear form pulled back through Johnson homomorphisms, and they explicitly determine the bilinear coefficients that govern in terms of and related constructs. Collectively, these results refine the understanding of finite-type invariants in low degree and illuminate the limitations of low-level Johnson filtrations to capture all homology spheres.

Abstract

We study the behaviour of the Casson invariant , its square, and Othsuki's second invariant as functions on the Johnson subgroup of the mapping class group. We show that since and are invariants that are morphisms on respectively the second and the third level of the Johnson filtration they never vanish on any level of this filtration. In contrast we prove that the invariant vanishes on the fifth level of the Johnson filtration, , and as a consequence we prove that, for instance, the Poincaré homology sphere does not admit any Heegaard splitting with gluing map an element in . Finally we determine a surgery formula for Othsuki's second invariant .
Paper Structure (11 sections, 29 theorems, 113 equations, 4 figures)

This paper contains 11 sections, 29 theorems, 113 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminary results
  3. Heegard splittings of integral homology spheres
  4. Finite type invariants
  5. The rational Torelli and Johnson Lie algebras
  6. Representation theory for $GL_g(\mathbb{Z})$
  7. The handlebody groups in $\mathfrak{m}_{g,1}(2)$
  8. Invariants and their trivialized 2-cocycles
  9. Cocycles of invariants of degree at most $2$
  10. Cocycles and restriction of invariants to the Johnson filtration.
  11. The bilinear form associated to $\lambda_2$

Key Result

Theorem 1.1

Figures (4)

  • Figure 1: Homology basis of $H_1(\Sigma_{g,1};\mathbb{Z})$
  • Figure 2: Trefoil knot $K$ and eight knot $L$ embedded in $\Sigma_{2,1}$
  • Figure 3: Trefoil knot embedded in $\Sigma_{2,1}$
  • Figure 4: Eight knot embedded in $\Sigma_{2,1}$

Theorems & Definitions (50)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1
  • Lemma 2.1
  • Proposition 2.1
  • proof
  • Corollary 2.1
  • proof
  • ...and 40 more