[Paper Review] Combinatorialization of spaces of nondegenerate spherical curves

Victor Goulart, Nicolau C. Saldanha|arXiv (Cornell University)|Oct 19, 2018

Topological and Geometric Data Analysis44 references4 citations

TL;DR

This paper introduces a combinatorial model for the homotopy type of spaces of nondegenerate spherical curves in $\mathbb{S}^n$ by associating each curve with an itinerary word in the symmetric group $S_{n+1} \setminus \{e\}$, which encodes the sequence of Schubert cells traversed by the Frenet frame. It constructs an abstract cell complex weakly homotopy equivalent to the space of curves with fixed initial and final frames, enabling computation of homotopy types via combinatorial data.

ABSTRACT

A parametric curve $γ$ of class $C^n$ on the $n$-sphere is said to be nondegenerate (or locally convex) when $\det\left(γ(t),γ'(t),\cdots,γ^{(n)}(t) ight)>0$ for all values of the parameter $t$. We orthogonalize this ordered basis to obtain the Frenet frame $\mathfrak{F}_γ$ of $γ$ assuming values in the orthogonal group $\operatorname{SO}_{n+1}$ (or its universal double cover, $\operatorname{Spin}_{n+1}$), which we decompose into Schubert or Bruhat cells. To each nondegenerate curve $γ$ we assign its itinerary: a word $w$ in the alphabet $S_{n+1}\smallsetminus\{e\}$ that encodes the succession of non open Schubert cells pierced by the complete flag of $\mathbb{R}^{n+1}$ spanned by the columns of $\mathfrak{F}_γ$. Without loss of generality, we can focus on nondegenerate curves with initial and final flags both fixed at the (non oriented) standard complete flag. For such curves, given a word $w$, the subspace of curves following the itinerary $w$ is a contractible globally collared topological submanifold of finite codimension. By a construction reminiscent of Poincaré duality, we define abstract cell complexes mapped into the original space of curves by weak homotopy equivalences. The gluing instructions come from a partial order in the set of words. The main aim of this construction is to attempt to determine the homotopy type of spaces of nondegenerate curves for $n>2$. The reader may want to contrast the present paper's combinatorial approach with the geometry-flavoured methods of previous works.

Motivation & Objective

• To determine the homotopy type of spaces of nondegenerate $C^n$ curves on the $n$-sphere $\mathbb{S}^n$ for $n > 2$, a problem previously approached geometrically.
• To develop a combinatorial framework that replaces differential-geometric methods with discrete data encoded in words over $S_{n+1} \setminus \{e\}$.
• To construct an abstract cell complex weakly homotopy equivalent to the space $\mathcal{L}_n(z)$ of nondegenerate curves with fixed initial and final Frenet frames in $\operatorname{Spin}_{n+1}$.
• To establish that each stratum $\mathcal{L}_n[w]$ of curves with a given itinerary word $w$ is a contractible, globally collared submanifold of finite codimension in $\mathcal{L}_n(q_w)$.
• To provide a systematic recipe for computing homotopy types using a partial order on words and Poincaré duality-inspired gluing instructions.

Proposed method

• Assign each nondegenerate curve $\gamma$ an itinerary $w = \operatorname{iti}(\gamma)$, a word in $S_{n+1} \setminus \{e\}$, encoding the sequence of non-open Schubert cells pierced by the complete flag spanned by the Frenet frame $\mathfrak{F}_\gamma$.
• Decompose the Frenet frame $\mathfrak{F}_\gamma$ into Schubert or Bruhat cells via the Bruhat decomposition of $\operatorname{GL}_{n+1}$, indexed by permutations in $S_{n+1}$.
• Define the dimension of a word $w = (\sigma_1, \dots, \sigma_\ell)$ as $\dim(w) = \sum_{i=1}^\ell (\operatorname{inv}(\sigma_i) - 1)$, which corresponds to the codimension of the stratum $\mathcal{L}_n[w]$.
• Construct an abstract cell complex by gluing cells indexed by words $w$ according to a partial order on $\mathbf{W}_n$, with gluing rules derived from the Bruhat order on $S_{n+1}$.
• Use the monodromy map $\mu_{z_0}$ from the space of $L^2$-regularized curves to $\operatorname{Spin}_{n+1}$ to show that monodromy subspaces are closed embedded submanifolds of codimension $n(n+1)/2$.
• Establish weak homotopy equivalence between the abstract cell complex and the original curve space $\mathcal{L}_n(z)$ via smoothness and surjectivity of the monodromy map, and use Hilbert manifold regularization to ensure differentiability.

Experimental results

Research questions

• RQ1 Can the homotopy type of the space of nondegenerate $C^n$ curves on $\mathbb{S}^n$ be determined combinatorially, independent of differential geometry?
• RQ2 How can the Frenet frame’s evolution through Schubert cells be encoded in a finite word over $S_{n+1} \setminus \{e\}$, and what does this itinerary reveal about the curve’s topology?
• RQ3 Is the subspace of curves with a fixed itinerary $w$ a contractible, globally collared submanifold of finite codimension in some $\mathcal{L}_n(q)$?
• RQ4 Can a weak homotopy equivalence be constructed between an abstract cell complex built from itinerary words and the space $\mathcal{L}_n(z)$ of curves with fixed initial and final frames?
• RQ5 What is the role of the monodromy map $\mu_{z_0}$ in relating the smooth structure of the curve space to the global topology of $\operatorname{Spin}_{n+1}$?

Key findings

• For each itinerary word $w \in \mathbf{W}_n$, the subspace $\mathcal{L}_n[w]$ of curves with that itinerary is a contractible, globally collared topological submanifold of $\mathcal{L}_n(q_w)$ of codimension $\dim(w)$.
• There exists a canonical map $w \mapsto q_w$ from words to elements of the Clifford group $\operatorname{CG}_{n+1}$, such that $\mathcal{L}_n[w] \subset \mathcal{L}_n(q_w)$.
• The monodromy map $\mu_{z_0}: \mathcal{L}_n^{[L^2]}(z_0; \cdot) \to \operatorname{Spin}_{n+1}$ is a surjective smooth submersion, implying that each monodromy subspace $\mathcal{L}_n^{[L^2]}(z_0; z_1)$ is a closed embedded submanifold of codimension $n(n+1)/2$.
• The inclusion of $C^n$-regularized curve spaces into $L^2$-regularized ones induces a homotopy equivalence of pairs, justifying the use of either topology for homotopy-theoretic purposes.
• The abstract cell complex constructed from itinerary words, with gluing rules based on the Bruhat order, is weakly homotopy equivalent to $\mathcal{L}_n(z)$, providing a combinatorial model for its homotopy type.
• The construction provides a systematic, combinatorial alternative to previous geometry-flavored approaches, enabling explicit computation of homotopy types for $n > 2$.