Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Homotopy types of diffeomorphism groups of polar Morse-Bott foliations on lens spaces, 1

Oleksandra Khokhliuk, Sergiy Maksymenko

Abstract

Let $T= S^1\times D^2$ be the solid torus, $\mathcal{F}$ the Morse-Bott foliation on $T$ into $2$-tori parallel to the boundary and one singular circle $S^1\times 0$, which is the central circle of the torus $T$, and $\mathcal{D}(\mathcal{F},\partial T)$ the group of diffeomorphisms of $T$ fixed on $\partial T$ and leaving each leaf of the foliation $\mathcal{F}$ invariant. We prove that $\mathcal{D}(\mathcal{F},\partial T)$ is contractible. Gluing two copies of $T$ by some diffeomorphism between their boundaries, we will get a lens space $L_{p,q}$ with a Morse-Bott foliation $\mathcal{F}_{p,q}$ obtained from $\mathcal{F}$ on each copy of $T$. We also compute the homotopy type of the group $\mathcal{D}(\mathcal{F}_{p,q})$ of diffeomorphisms of $L_{p,q}$ leaving invariant each leaf of $\mathcal{F}_{p,q}$.

Homotopy types of diffeomorphism groups of polar Morse-Bott foliations on lens spaces, 1

Abstract

Let be the solid torus, the Morse-Bott foliation on into -tori parallel to the boundary and one singular circle , which is the central circle of the torus , and the group of diffeomorphisms of fixed on and leaving each leaf of the foliation invariant. We prove that is contractible. Gluing two copies of by some diffeomorphism between their boundaries, we will get a lens space with a Morse-Bott foliation obtained from on each copy of . We also compute the homotopy type of the group of diffeomorphisms of leaving invariant each leaf of .
Paper Structure (27 sections, 16 theorems, 80 equations, 2 figures)

This paper contains 27 sections, 16 theorems, 80 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main result
  3. Preliminaries
  4. Deformations
  5. Notation for several diffeomorphism groups
  6. Partitions
  7. Tangent maps along fibers
  8. Homogeneous functions on vector bundles
  9. Shifts along orbits of flows
  10. Linearization theorem
  11. Main result
  12. Trivial vector bundles of rank $1$
  13. Product foliations and loop spaces
  14. Vector bundles over $1$-dimensional manifolds
  15. Vector bundles over the circle
  16. ...and 12 more sections

Key Result

Theorem 1.1.1

The group $\mathcal{D}_{\mathrm{fix}}(\mathcal{F},T^2)$ is weakly contractible, that is weakly homotopy equivalent to a point (i.e. all its homotopy groups vanish).

Figures (2)

  • Figure 5.1: Foliation $\mathcal{F}$
  • Figure 5.2: Covering spaces

Theorems & Definitions (33)

  • Theorem 1.1.1
  • Theorem 1.1.2
  • Theorem 1.1.3
  • Remark 1.1.4
  • Lemma 2.5.1: c.f. Maksymenko:TA:2003, KhokhliukMaksymenko:2022
  • proof
  • Lemma 2.6.1: Maksymenko:TA:2003
  • Remark 3.1.1
  • Theorem 3.1.2: c.f. KhokhliukMaksymenko:2022
  • proof
  • ...and 23 more