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On automorphisms of high-dimensional solid tori

Mauricio Bustamante, Oscar Randal-Williams

Abstract

We study the infinite generation in the homotopy groups of the group of diffeomorphisms of $S^1 \times D^{2n-1}$, for $2n \geq 6$, in a range of degrees up to $n-2$. Our analysis relies on understanding the homotopy fibre of a linearisation map from the plus-construction of the classifying space of certain space of self-embeddings of stabilisations of this manifold to a form of Hermitian $K$-theory of the integral group ring of $π_1(S^1)$. We also show that these homotopy groups vanish rationally.

On automorphisms of high-dimensional solid tori

Abstract

We study the infinite generation in the homotopy groups of the group of diffeomorphisms of , for , in a range of degrees up to . Our analysis relies on understanding the homotopy fibre of a linearisation map from the plus-construction of the classifying space of certain space of self-embeddings of stabilisations of this manifold to a form of Hermitian -theory of the integral group ring of . We also show that these homotopy groups vanish rationally.

Paper Structure

This paper contains 40 sections, 41 theorems, 263 equations, 1 figure.

Table of Contents

  1. Introduction and statement of results
  2. Manifolds and quadratic modules
  3. Manifolds
  4. Quadratic modules
  5. Homotopy automorphisms of $X_g$
  6. A fibration sequence
  7. The mapping class group
  8. Higher homotopy groups
  9. The Weiss sequence, disjunction, and surgery
  10. The Weiss fibre sequence
  11. Disjunction and surgery
  12. Self-embeddings of $X_g$
  13. Realising automorphisms of a quadratic module by self-embeddings
  14. Realising automorphisms of a quadratic module by diffeomorphisms
  15. The structure set
  16. ...and 25 more sections

Key Result

Theorem A

For $n \geq 3$, all primes $p$, and $0 < k<\min(2p-3, n-2)$, the groups $\pi_k(B\mathop{\mathrm{Diff}}\nolimits_{\partial}(S^1\times D^{2n-1}))$ are finitely-generated when localised at $p$. Furthermore if $2p-3 < n-2$ then there is a map which is injective and whose cokernel is finitely-generated after localisation at $p$. When $p=2$ the cokernel is finitely-generated even before localisation.

Figures (1)

  • Figure 1: The manifolds $X_g$ (left) and $W_{g,1}$ (right) for $g=2$. Note that $W_{g,1}\hookrightarrow X_g\hookrightarrow W_{g,1}$.

Theorems & Definitions (96)

  • Theorem A
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem B
  • Remark 1.5
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • ...and 86 more