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Surgery equivalence relations for 3-manifolds

Gwenael Massuyeau

Abstract

By classical results of Rochlin, Thom, Wallace and Lickorish, it is well-known that any two 3-manifolds (with diffeomorphic boundaries) are related one to the other by surgery operations. Yet, by restricting the type of the surgeries, one can define several families of non-trivial equivalence relations on the sets of (diffeomorphism classes of) 3-manifolds. In this expository paper, which is based on lectures given at the school ``Winter Braids XI'' (Dijon, December 2021), we explain how certain filtrations of mapping class groups of surfaces enter into the definitions and the mutual comparison of these surgery equivalence relations. We also survey the ways in which concrete invariants of 3-manifolds (such as finite-type invariants) can be used to characterize such relations.

Surgery equivalence relations for 3-manifolds

Abstract

By classical results of Rochlin, Thom, Wallace and Lickorish, it is well-known that any two 3-manifolds (with diffeomorphic boundaries) are related one to the other by surgery operations. Yet, by restricting the type of the surgeries, one can define several families of non-trivial equivalence relations on the sets of (diffeomorphism classes of) 3-manifolds. In this expository paper, which is based on lectures given at the school ``Winter Braids XI'' (Dijon, December 2021), we explain how certain filtrations of mapping class groups of surfaces enter into the definitions and the mutual comparison of these surgery equivalence relations. We also survey the ways in which concrete invariants of 3-manifolds (such as finite-type invariants) can be used to characterize such relations.
Paper Structure (17 sections, 35 theorems, 94 equations, 4 figures)

This paper contains 17 sections, 35 theorems, 94 equations, 4 figures.

Table of Contents

  1. Basics about $3$-manifolds and mapping class groups
  2. Surgeries and handle decompositions
  3. Mapping class groups of surfaces
  4. Triviality of $\Omega_3$
  5. Surgery equivalence relations: definitions and first properties
  6. Torelli groups of surfaces
  7. Torelli twists in $3$-manifolds
  8. Filtrations on the Torelli groups
  9. Stronger surgeries in $3$-manifolds
  10. Clasper calculus
  11. Other kinds of surgeries
  12. Surgery equivalence relations: their characterization
  13. Two case studies to consider
  14. Characterization of the Torelli--equivalence
  15. Characterization of $J_k$ and $Y_k$ at low $k$ for closed manifolds
  16. ...and 2 more sections

Key Result

Lemma 1.6

Let $g\in {\mathbb N}$. The (oriented) diffeomorphism type of $M_f := {H_g \cup_f (-H_g)}$ only depends on the isotopy class of $f$.

Figures (4)

  • Figure 1: A knot $K$ (black) in its regular neighborhood $\hbox{N}(K)$, together with the meridian (red) and a parallel (blue)
  • Figure 2: Dehn's generators in genus $1$ and $2$
  • Figure 3: Dehn's generators in genus $g>2$
  • Figure 4: A $Y$-graph and the associated framed link

Theorems & Definitions (88)

  • Example 1.1
  • Remark 1.2
  • Definition 1.3
  • Example 1.4
  • Example 1.5
  • Lemma 1.6
  • proof
  • Example 1.7
  • Theorem 1.8: Dehn 1938
  • Remark 1.9
  • ...and 78 more