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Diffeomorphism groups of Morse-Bott foliation on the solid Klein bottle by Klein bottles parallel to the boundary

Sergiy Maksymenko

Abstract

Let $\mathcal{G}$ be a Morse-Bott foliation on the solid Klein bottle $\mathbf{K}$ into $2$-dimensional Klein bottles parallel to the boundary and one singular circle $S^1$. Let also $S^1\widetilde{\times}S^2$ be the twisted bundle over $S^1$ which is a union of two solid Klein bottles $\mathbf{K}_0$ and $\mathbf{K}_1$ with common boundary $K$. Then the above foliations $\mathcal{G}$ on both $\mathbf{K}_0$ and $\mathbf{K}_1$ gives a foliation $\mathcal{G}'$ on $S^1\widetilde{\times}S^2$ into parallel Klein bottles and two singluar circles. The paper computes the homotopy types of groups of foliated (sending leaves to leaves) and leaf preserving diffeomorphisms for foliations $\mathcal{G}$ and $\mathcal{G}'$.

Diffeomorphism groups of Morse-Bott foliation on the solid Klein bottle by Klein bottles parallel to the boundary

Abstract

Let be a Morse-Bott foliation on the solid Klein bottle into -dimensional Klein bottles parallel to the boundary and one singular circle . Let also be the twisted bundle over which is a union of two solid Klein bottles and with common boundary . Then the above foliations on both and gives a foliation on into parallel Klein bottles and two singluar circles. The paper computes the homotopy types of groups of foliated (sending leaves to leaves) and leaf preserving diffeomorphisms for foliations and .
Paper Structure (7 sections, 7 theorems, 12 equations, 1 figure)

This paper contains 7 sections, 7 theorems, 12 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Twisted bundles over the circle
  3. Universal cover of the solid Klein bottle $\mathbf{K}$
  4. Universal cover of $\mathbf{K}\setminus K_{0}$
  5. Liftings of diffemorphisms
  6. Proof of Theorem \ref{['th:homtype:Kfol']}
  7. Proof of Theorem \ref{['th:homtype:S2nxS1']}

Key Result

Theorem 1.1

The group $\mathcal{D}^{lp}(\mathcal{F},\partial\mathbf{T})$ is weakly contractible.

Figures (1)

  • Figure 2.1: Universal coverings of $\mathbf{K}$ and $\mathbf{K}\setminus K_{0}$

Theorems & Definitions (10)

  • Theorem 1.1: KhokhliukMaksymenko:lens:2022
  • Theorem 1.2: Maksymenko:lens:2023
  • Lemma 1.3
  • proof : History of proof
  • Theorem 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Lemma 2.4
  • proof
  • Remark 3.2