Homotopy type of spaces of locally convex curves in the sphere S^3
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2022
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866912812533547008 |
|---|---|
| author | Alves, Emília Goulart, Victor Saldanha, Nicolau C. |
| author_facet | Alves, Emília Goulart, Victor Saldanha, Nicolau C. |
| contents | Locally convex (or nondegenerate) curves in the sphere $S^n$ have been studied for several reasons, including the study of linear ordinary differential equations of order $n+1$. Taking Frenet frames allows us to obtain corresponding curves $Γ$ in the group $Spin_{n+1}$. Let $L_n(z_0;z_1)$ be the space of such curves $Γ$ with prescribed endpoints $Γ(0) = z_0$, $Γ(1) = z_1$. The aim of this paper is to determine the homotopy type of the spaces $L_3(z_0;z_1)$ for all $z_0, z_1 \in Spin_4$. As a corollary, we obtain the homotopy type of the space of closed locally convex curves in either $S^3$ or $P^3$.
There are many previous papers addressing related questions. An early paper solves the corresponding problem for curves in $S^2$. Another previous result (with B. Shapiro) reduces the problem to $z_0 = 1$ and $z_1 \in Quat_4$ where $Quat_4 \subset Spin_4$ is a finite group of order $16$. A more recent paper shows that for $z_1 \in Quat_4 \smallsetminus Z(Quat_4)$ we have a homotopy equivalence $L_3(1;z_1) \approx ΩSpin_4$. In this paper we compute the homotopy type of $L_3(1;z_1)$ for $z_1 \in Z(Quat_4)$: it is equivalent to the wedge of $ΩSpin_4$ with an infinite countable family of spheres (as for the case $n = 2$).
The structure of the proof can be compared to that of the case $n = 2$ but some of the steps require the creation of new theories, involving algebra and combinatorics. We construct explicit subsets $Y \subset L_n(z_0;z_1)$ for which the inclusion $Y \subset ΩSpin_{n+1}(z_0;z_1)$ is a homotopy equivalence. For $n = 2$, there is a simple geometric description of $Y$; for $n = 3$, the far less natural construction is based on the theory of itineraries of such curves. The itinerary of a curve in $L_n(1;z_1)$ is a finite word in the alphabet $S_{n+1} \smallsetminus \{e\}$ of nontrivial permutations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2205_10928 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Homotopy type of spaces of locally convex curves in the sphere S^3 Alves, Emília Goulart, Victor Saldanha, Nicolau C. Geometric Topology Algebraic Topology 53C42, 34B05, 55P15, 57N20, 58B05 Locally convex (or nondegenerate) curves in the sphere $S^n$ have been studied for several reasons, including the study of linear ordinary differential equations of order $n+1$. Taking Frenet frames allows us to obtain corresponding curves $Γ$ in the group $Spin_{n+1}$. Let $L_n(z_0;z_1)$ be the space of such curves $Γ$ with prescribed endpoints $Γ(0) = z_0$, $Γ(1) = z_1$. The aim of this paper is to determine the homotopy type of the spaces $L_3(z_0;z_1)$ for all $z_0, z_1 \in Spin_4$. As a corollary, we obtain the homotopy type of the space of closed locally convex curves in either $S^3$ or $P^3$. There are many previous papers addressing related questions. An early paper solves the corresponding problem for curves in $S^2$. Another previous result (with B. Shapiro) reduces the problem to $z_0 = 1$ and $z_1 \in Quat_4$ where $Quat_4 \subset Spin_4$ is a finite group of order $16$. A more recent paper shows that for $z_1 \in Quat_4 \smallsetminus Z(Quat_4)$ we have a homotopy equivalence $L_3(1;z_1) \approx ΩSpin_4$. In this paper we compute the homotopy type of $L_3(1;z_1)$ for $z_1 \in Z(Quat_4)$: it is equivalent to the wedge of $ΩSpin_4$ with an infinite countable family of spheres (as for the case $n = 2$). The structure of the proof can be compared to that of the case $n = 2$ but some of the steps require the creation of new theories, involving algebra and combinatorics. We construct explicit subsets $Y \subset L_n(z_0;z_1)$ for which the inclusion $Y \subset ΩSpin_{n+1}(z_0;z_1)$ is a homotopy equivalence. For $n = 2$, there is a simple geometric description of $Y$; for $n = 3$, the far less natural construction is based on the theory of itineraries of such curves. The itinerary of a curve in $L_n(1;z_1)$ is a finite word in the alphabet $S_{n+1} \smallsetminus \{e\}$ of nontrivial permutations. |
| title | Homotopy type of spaces of locally convex curves in the sphere S^3 |
| topic | Geometric Topology Algebraic Topology 53C42, 34B05, 55P15, 57N20, 58B05 |
| url | https://arxiv.org/abs/2205.10928 |