Homotopy type of spaces of locally convex curves in the sphere S^3

TL;DR

This work determines the homotopy type of spaces of locally convex curves in ${\mathbb S}^3$ by translating the problem to Frenet frames in ${\rm Spin}_4$ and exploiting Bruhat cell stratifications and itinerary combinatorics. For noncentral endpoint data $z\in Z(\operatorname{Quat}_4)$, the spaces ${\cal L}_3(1;z)$ are weakly equivalent to loop spaces $\Omega{\rm Spin}_4$, while for central elements the homotopy type is a bouquet of $\Omega{\rm Spin}_4$ with infinitely many spheres of increasing dimension; the paper gives precise sphere multiplicities in each center case. Central to the analysis are the construction of contractible subsets ${\cal M}_{\mu_0,\mu_1}$ corresponding to itineraries, the study of their closures, and the development of explicit maps ${h}_{\mu_0,\mu_1}$ whose intersections with ${\cal M}_{\mu_0,\mu_1}$ are controlled and transversal. The results yield a detailed description of the homotopy type of spaces of closed locally convex curves in ${\mathbb S}^3$ or ${\mathbb P}^3$, with implications for associated linear ODE theories and a robust combinatorial framework based on itineraries and parity-alternating permutations.

Abstract

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.

Figures (5)

• Figure 1: The cells $c_{[aba]}$ and $c_{[bcb]}$.
• Figure 2: The cells $c_{[abc]}$ and $c_{[cba]}$.
• Figure 3: A transversal section to ${\cal L}_3[[bacb]]$. Since ${\cal L}_3[[bacb]] \subset {\cal L}_3$ has codimension $3$, the X figure should be seen in a space of dimension $3$. The thicker lines show ${\cal L}_3[[aba]]$ and ${\cal L}_3[[bcb]]$, which are also contained in ${\cal M}$. The thinner lines are smooth continuations, but are not in ${\cal M}$.
• Figure 4: A transversal section to ${\cal L}_3[[bacb]]$. The $2$-sphere is drawn like the Earth in a Mercator map. The north and south poles are $\pm e_1$, with $-e_1$ above and $e_1$ below. The equator is a horizontal line in the middle (not drawn) and corresponds to $x_1 = 0$, with the points $e_2$, $-e_3$, $-e_2$ and $e_3$ reading from left to right. Zero sets of $q_{[ab]}$, $q_{[ba]}$, $q_{[ac]}$, $q_{[bc]}$ and $q_{[cb]}$ are shown in black, red, green, blue and magenta, respectively. The polynomial $q_{[ac]}$ is a square and the green lines are actually double. The one at the center has the effect of swapping two pairs $ac \leftrightarrow ca$; the one appearing twice at both ends of the figure detects the coincidence of complex roots and therefore swaps nothing in terms of itineraries or real roots. This and similar examples are joint work with Boris Shapiro. The figure was produced with the help of the Maple software.
• Figure 5: A valid boundary for $[bacb]$.

Theorems & Definitions (64)

• Theorem 1
• Remark 1.2
• Remark 1.3
• Theorem 2
• Remark 2.2
• Remark 2.3
• Proposition 2.5
• Remark 3.2
• Example 4.1
• Remark 4.2