A CW complex homotopy equivalent to spaces of locally convex curves

Victor Goulart; Nicolau C. Saldanha

paper

A CW complex homotopy equivalent to spaces of locally convex curves

arXiv:2112.14539

Abstract

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 $S p i n_{n + 1}$ ; $Π : S p i n_{n + 1} \to F l a g_{n + 1}$ 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 $Γ : [0, 1] \to S p i n_{n + 1}$ 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 $S_{n + 1} ∖ {e}$ . 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 $Ω S p i n_{n + 1}$ . For all $n \geq 2$ , all connected components of $L_{n}$ are simply connected.

56 pages, 15 figures. Updated references. Text overlap with arXiv:1810.08632