Homotopy types of diffeomorphism groups of polar Morse-Bott foliations on lens spaces, 1
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| Format: | Preprint |
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2022
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| author | Khokhliuk, Oleksandra Maksymenko, Sergiy |
| author_facet | Khokhliuk, Oleksandra Maksymenko, Sergiy |
| contents | 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 $\mathcal{F}_{p,q}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2210_11043 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Homotopy types of diffeomorphism groups of polar Morse-Bott foliations on lens spaces, 1 Khokhliuk, Oleksandra Maksymenko, Sergiy Algebraic Topology Complex Variables Differential Geometry Dynamical Systems Geometric Topology 57R30, 57T20 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 $\mathcal{F}_{p,q}$. |
| title | Homotopy types of diffeomorphism groups of polar Morse-Bott foliations on lens spaces, 1 |
| topic | Algebraic Topology Complex Variables Differential Geometry Dynamical Systems Geometric Topology 57R30, 57T20 |
| url | https://arxiv.org/abs/2210.11043 |