A CW complex homotopy equivalent to spaces of locally convex curves

TL;DR

This work constructs a CW complex ${\\cal D}_n$ that is homotopy equivalent to the space ${\\cal L}_n$ of locally convex curves in ${\\rm Spin}_{n+1}$, with cells indexed by words in a combinatorial alphabet tied to Bruhat cells. By organizing strata via two partial orders and using lower-set filtrations, the authors obtain explicit 1- and 2-skeletons, prove simple connectivity of all components, and define open subspaces ${\\cal Y}_n$ that model loop-space behavior, yielding a weak equivalence to ${\\Omega} \\ Spin_{n+1}$ for most endpoints. They show most connected components are captured by ${\\cal Y}_n$, while central-endpoints require more delicate treatment, with detailed results for the case $n=3$ (the Alves-Goulart-Saldanha description of ${\\cal L}_3$). The work links the topology of locally convex curves to combinatorial stratification, with implications for differential operators and symplectic structures via the Adler-Gelfand-Dickey framework. Overall, it provides a robust combinatorial toolkit to approximate and understand the homotopy types of spaces of locally convex curves for higher $n$.

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 $Spin_{n+1}$; $Π: Spin_{n+1} \to Flag_{n+1}$ 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.

Figures (15)

• Figure 1: A family of curves in ${\cal L}_2$. The equator is dashed and the fat dot indicates $e_1$. The vector $e_2$ is at the right. Here we use the simplified notation for words in ${\mathbf W}_{2}$: for instance, $abab$ denotes $(a,b,a,b)$ and $[aba]$ denotes $(aba)$.
• Figure 2: The $\blacktriangleleft$-pairs in $S_{\operatorname{PP}} \subset S_5$.
• Figure 3: The lower set ${\mathbf I}([aba])$.
• Figure 4: The cells $c_{[aba]}$ and $c_{[bcb]}$.
• Figure 5: Examples of edges of ${\cal D}_n$.

Theorems & Definitions (89)

• Theorem 1
• Theorem 2
• Theorem 3
• Corollary 1.1
• Example 2.1
• Lemma 2.2
• Remark 2.3
• proof : Proof of Lemma \ref{['lemma:PAhat']}
• Remark 2.4
• Lemma 2.5