A CW complex homotopy equivalent to spaces of locally convex curves
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2021
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| _version_ | 1866915756512378880 |
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| author | Goulart, Victor Saldanha, Nicolau C. |
| author_facet | Goulart, Victor Saldanha, Nicolau C. |
| contents | Locally convex curves in the sphere $S^n$ have been studied for several reasons, including the study of linear ordinary differential equations. Taking Frenet frames obtains corresponding curves $Γ$ in the group $Spin_{n+1}$; $Π: Spin_{n+1} \to Flag_{n+1}$ is the universal cover of the space of flags. Determining the homotopy type of spaces of such curves $Γ$ with prescribed initial and final points appears to be a hard problem. We may focus on $L_n$, the space of locally convex curves $Γ: [0,1] \to Spin_{n+1}$ with $Γ(0) = 1$, $Π(Γ(1)) = Π(1)$. Convex curves form a contractible connected component of $L_n$; there are $2^{n+1}$ other components, one for each endpoint. The homotopy type of $L_n$ has so far been determined only for $n=2$. This paper is a step towards solving the problem for larger values of $n$.
The itinerary of $Γ$ belongs to $W_n$, the set of finite words in the alphabet $S_{n+1} \setminus \{e\}$. The itinerary of a curve lists the non open Bruhat cells crossed. Itineraries stratify the space $L_n$. We construct a CW complex $D_n$ which is a kind of dual of $L_n$ under this stratification: the construction is similar to Poincaré duality. The CW complex $D_n$ is homotopy equivalent to $L_n$. The cells of $D_n$ are naturally labeled by words in $W_n$; $D_n$ is locally finite. Explicit glueing instructions are described for lower dimensions.
We describe an open subset $Y_n \subset L_n$, a union of strata of $L_n$. In each non convex component of $L_n$, the intersection with $Y_n$ is connected and dense. Most connected components of $L_n$ are contained in $Y_n$. For $n > 3$, in the other components the complement of $Y_n$ has codimension at least $2$. The set $Y_n$ is homotopy equivalent to the disjoint union of $2^{n+1}$ copies of $ΩSpin_{n+1}$. For all $n \ge 2$, all connected components of $L_n$ are simply connected. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2112_14539 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | A CW complex homotopy equivalent to spaces of locally convex curves Goulart, Victor Saldanha, Nicolau C. Geometric Topology Algebraic Topology 53C42, 34B05, 55P15, 57N20, 58B05 Locally convex curves in the sphere $S^n$ have been studied for several reasons, including the study of linear ordinary differential equations. Taking Frenet frames obtains corresponding curves $Γ$ in the group $Spin_{n+1}$; $Π: Spin_{n+1} \to Flag_{n+1}$ is the universal cover of the space of flags. Determining the homotopy type of spaces of such curves $Γ$ with prescribed initial and final points appears to be a hard problem. We may focus on $L_n$, the space of locally convex curves $Γ: [0,1] \to Spin_{n+1}$ with $Γ(0) = 1$, $Π(Γ(1)) = Π(1)$. Convex curves form a contractible connected component of $L_n$; there are $2^{n+1}$ other components, one for each endpoint. The homotopy type of $L_n$ has so far been determined only for $n=2$. This paper is a step towards solving the problem for larger values of $n$. The itinerary of $Γ$ belongs to $W_n$, the set of finite words in the alphabet $S_{n+1} \setminus \{e\}$. The itinerary of a curve lists the non open Bruhat cells crossed. Itineraries stratify the space $L_n$. We construct a CW complex $D_n$ which is a kind of dual of $L_n$ under this stratification: the construction is similar to Poincaré duality. The CW complex $D_n$ is homotopy equivalent to $L_n$. The cells of $D_n$ are naturally labeled by words in $W_n$; $D_n$ is locally finite. Explicit glueing instructions are described for lower dimensions. We describe an open subset $Y_n \subset L_n$, a union of strata of $L_n$. In each non convex component of $L_n$, the intersection with $Y_n$ is connected and dense. Most connected components of $L_n$ are contained in $Y_n$. For $n > 3$, in the other components the complement of $Y_n$ has codimension at least $2$. The set $Y_n$ is homotopy equivalent to the disjoint union of $2^{n+1}$ copies of $ΩSpin_{n+1}$. For all $n \ge 2$, all connected components of $L_n$ are simply connected. |
| title | A CW complex homotopy equivalent to spaces of locally convex curves |
| topic | Geometric Topology Algebraic Topology 53C42, 34B05, 55P15, 57N20, 58B05 |
| url | https://arxiv.org/abs/2112.14539 |