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Embedding calculus for surfaces

Manuel Krannich, Alexander Kupers

Abstract

We prove convergence of Goodwillie-Weiss' embedding calculus for spaces of embeddings into a manifold of dimension at most two, so in particular for diffeomorphisms between surfaces. We also relate the Johnson filtration of the mapping class group of a surface to a certain filtration arising from embedding calculus.

Embedding calculus for surfaces

Abstract

We prove convergence of Goodwillie-Weiss' embedding calculus for spaces of embeddings into a manifold of dimension at most two, so in particular for diffeomorphisms between surfaces. We also relate the Johnson filtration of the mapping class group of a surface to a certain filtration arising from embedding calculus.

Paper Structure

This paper contains 23 sections, 17 theorems, 65 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Convergence in low dimensions
  3. Embedding calculus and the Johnson filtration
  4. Generalities on spaces of embeddings and embedding calculus
  5. Spaces of embeddings and maps
  6. Manifold calculus
  7. Manifold calculus in terms of presheaves
  8. A smaller model
  9. Two properties of manifold calculus
  10. Properties of embedding calculus
  11. Thickened embeddings
  12. Lifting along covering maps
  13. Adding a collar to the source
  14. Taking disjoint unions
  15. Convergence in low dimensions
  16. ...and 8 more sections

Key Result

Theorem 1

For compact manifolds $M$ and $N$ with $\dim(N)\le 2$, the map is a weak homotopy equivalence for any embedding $e_\partial\colon \partial M\hookrightarrow \partial N$.

Figures (7)

  • Figure 1: The surface $P$. The original surface $N$ is the region within the dotted circle.
  • Figure 2: The surface $R$.
  • Figure 3: The complement of an open disc in Möbius strip. The orange copy of $I \times [0,1]$ differs up to isotopy equivalence from ${\rm Mo}\backslash {\rm int}(D^2)$ only in hatched region which is diffeomorphic to $I \times [0,1) \sqcup I \times [0,1)$.
  • Figure 4: The triads $J,H_0 \subset D^2$. Here $H$ the union of $J$ and $H_0$, and $H'_0 \subset H_0$ is the component to the right of $J$.
  • Figure 5: $M=\Sigma_{0,4}\,\natural\,(\Sigma_{1,1})^{\natural \ul{2}}$ with subtriad $P = \ul{2} \times S^1 \times [0,1]\subset M$ whose complement has genus $0$ and $8$ boundary components.
  • ...and 2 more figures

Theorems & Definitions (40)

  • Theorem 1
  • Corollary 2
  • Remark \oldthetheorem
  • Example \oldthetheorem: Embedding calculus
  • Remark \oldthetheorem
  • Remark \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • Remark \oldthetheorem
  • Remark \oldthetheorem
  • ...and 30 more