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

TL;DR

This paper investigates the homotopy types of diffeomorphism groups of Morse-Bott foliations on lens spaces, establishing contractibility results and computing their homotopy groups.

Contribution

It proves the contractibility of the diffeomorphism group fixing the foliation on a solid torus and determines the homotopy type of such groups on lens spaces.

Findings

The diffeomorphism group fixing the foliation on the solid torus is contractible.

Homotopy types of diffeomorphism groups on lens spaces are explicitly computed.

Provides new insights into the topology of foliated lens spaces.

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 $\mathcal{F}_{p,q}$.