A CW complex homotopy equivalent to spaces of locally convex curves

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 $\Gamma$ in the group $Spin_{n+1}$; $\Pi: Spin_{n+1} \to Flag_{n+1}$ is the universal cover of the space of flags. Determining the homotopy type of spaces of such curves $\Gamma$ with prescribed initial and final points appears to be a hard problem. We may focus on $L_n$, the space of locally convex curves $\Gamma: [0,1] \to Spin_{n+1}$ with $\Gamma(0) = 1$, $\Pi(\Gamma(1)) = \Pi(1)$. 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 $\Gamma$ 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 Poincar\'e 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 $\Omega Spin_{n+1}$. For all $n \ge 2$, all connected components of $L_n$ are simply connected.