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Stability of concordance embeddings

Thomas Goodwillie, Manuel Krannich, Alexander Kupers

TL;DR

This work proves a multirelative stability theorem for spaces of smooth concordance embeddings, giving precise connectivity bounds that depend on the handle dimensions of boundary intersections and submanifold removals. The authors develop a robust framework using cubical diagrams, stabilization and scanning maps, the delooping trick, and multirelative Blakers–Massey to propagate connectivity through point, disc, and general handle cases. They then translate these results into applications for spaces of concordance diffeomorphisms and homeomorphisms, yielding stability and rational-stable-range consequences via smoothing theory and tangential-type data. The findings illuminate how concordance spaces stabilize under manifold modifications and provide tools for computing their (rational) homotopy types and mapping-class-group structures.

Abstract

We prove a stability theorem for spaces of smooth concordance embeddings. From it we derive various applications to spaces of concordance diffeomorphisms and homeomorphisms.

Stability of concordance embeddings

TL;DR

This work proves a multirelative stability theorem for spaces of smooth concordance embeddings, giving precise connectivity bounds that depend on the handle dimensions of boundary intersections and submanifold removals. The authors develop a robust framework using cubical diagrams, stabilization and scanning maps, the delooping trick, and multirelative Blakers–Massey to propagate connectivity through point, disc, and general handle cases. They then translate these results into applications for spaces of concordance diffeomorphisms and homeomorphisms, yielding stability and rational-stable-range consequences via smoothing theory and tangential-type data. The findings illuminate how concordance spaces stabilize under manifold modifications and provide tools for computing their (rational) homotopy types and mapping-class-group structures.

Abstract

We prove a stability theorem for spaces of smooth concordance embeddings. From it we derive various applications to spaces of concordance diffeomorphisms and homeomorphisms.
Paper Structure (22 sections, 32 theorems, 80 equations, 6 figures)

This paper contains 22 sections, 32 theorems, 80 equations, 6 figures.

Table of Contents

  1. Introduction
  2. The multirelative stability theorem and some preliminaries
  3. Cubical diagrams
  4. The stabilisation map
  5. Statement of the main theorem
  6. Collars and tubular neighborhoods
  7. Previous multiple disjunction results
  8. The delooping trick and scanning
  9. Concordance maps and immersions
  10. The stabilisation and scanning maps for concordance maps and immersions
  11. The proof of the multirelative stability theorem
  12. The case of a point
  13. Digression
  14. The space $\mathrm{CE}^{\{t_1,t_2\}}(\ast, M)$
  15. A variation of Step \ref{['step:blakers-massey']}
  16. ...and 7 more sections

Key Result

Theorem A

If the handle dimension $p$ of $\partial M\cap P\subset P$ satisfies $p\le d-3$, then the map is $(2d-p-5)$-connected.

Figures (6)

  • Figure 1: The stabilisation map.
  • Figure 2: The decomposition $J \times I = D_1 \cup D_2$. The red arcs indicate the parametrisation \ref{['equ:polar-coordinates']}: the semicircle is parametrised by fixing $r=0$ and taking $\theta\in [0,\pi]$ starting with $\theta = 0$ on the left, and the radial segments are parametrised by fixing $\theta \in [0,\pi]$ and taking $r\in [0,1]$ starting with $r=0$ at the semicircle. The map $\sigma(e)$ is given by the identity on $D_2$, and by $e$ on each radial segment in $D_1$.
  • Figure 3: The subspaces of $M$ appearing in the delooping trick.
  • Figure 4: An element $e$ of $\mathrm{CE}^A(*,M)$. The compact submanifold $A \subset I$ is indicated in thick red.
  • Figure 5: The restriction of $\mathrm{pr}_{J \times I} \circ \sigma(f)$ to the indicated interval changes the radial coordinate (in $[0,1]$) but not the angle (in $[0,\pi]$). In particular, the compositions with $\mathrm{pr}_{[0,\pi]} \circ \Lambda'^{-1} \circ \mathrm{pr}_{J \times I}$ of $\sigma(f)$ and $\sigma(\mathrm{inc})$ agree.
  • ...and 1 more figures

Theorems & Definitions (63)

  • Theorem A
  • Remark 1.1
  • Corollary B
  • Example
  • Corollary C
  • Remark 1.2
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Corollary 2.3
  • ...and 53 more