[Paper Review] Stratification by itineraries of spaces of locally convex curves

Victor Goulart, Nicolau C. Saldanha|arXiv (Cornell University)|Jul 2, 2019

Homotopy and Cohomology in Algebraic Topology31 references5 citations

TL;DR

This paper introduces a stratification of spaces of locally convex curves in $Spin_{n+1}$ using 'itineraries'—words encoding the sequence of non-open Bruhat cells traversed. It proves each stratum is a contractible submanifold of finite codimension, enabling construction of weak homotopy equivalences to CW complexes, and resolves a conjecture by Shapiro and Shapiro on the structure of such curve spaces.

ABSTRACT

Locally convex (or nondegenerate) curves in the sphere (or projective space) have been studied for several reasons, including the study of linear ordinary differential equations. Taking Frenet frames allows us to translate such curves into corresponding curves in the flag space, the orthogonal group or its cover $Spin_{n+1}$. Determining the homotopy type of the space of such closed curves or, more generally, of spaces of such curves with prescribed initial and final jets appears to be a hard problem, which has been solved for $n=2$ but otherwise remains open. This paper is a step towards solving the problem for larger values of $n$. In the process, we prove a related conjecture of B. Shapiro and M. Shapiro. We define the itinerary of a locally convex curve $\Gamma:[0,1] o Spin_{n+1}$ as a word $w$ in the alphabet of non-trivial permutations. This word encodes the succession of non-open Bruhat cells of $Spin_{n+1}$ pierced by $\Gamma$. We prove that, for each word $w$, the subspace of curves of itinerary $w$ is an embedded contractible (topological) submanifold of finite codimension, thus defining a stratification of the space of curves. We show how to obtain explicit (topologically) transversal sections for each of these submanifolds. We study both a space of curves with minimum regularity hypotheses, where only topological transversality applies, and spaces of sufficiently regular curves. In both cases we study the neighboring relation between strata. This is an important step in the construction of CW cell complexes mapped into the original space of curves by weak homotopy equivalences. Our stratification is not as nice as might be desired, lacking for instance the Whitney property. The differentiability class of the curves affects some properties of the stratification. The necessary ingredients for the construction of a dual CW complex are proved.

Motivation & Objective

• To understand the homotopy type of spaces of closed, locally convex curves in $Spin_{n+1}$, a problem open for $n > 2$.
• To define and analyze a stratification of such curve spaces based on the sequence of Bruhat cells traversed, termed 'itineraries'.
• To prove that each stratum defined by a fixed itinerary is a contractible topological submanifold of finite codimension.
• To establish conditions under which explicit transversal sections to these strata can be constructed.
• To provide foundational tools for constructing dual CW complexes weakly homotopy equivalent to the original curve space.

Proposed method

• Define the itinerary of a curve $\Gamma: [0,1] \to Spin_{n+1}$ as a word in non-trivial permutations, encoding the succession of non-open Bruhat cells it crosses.
• Prove that for each itinerary word $w$, the set of curves with itinerary $w$ forms a topological submanifold of finite codimension in the space of curves.
• Show that each such submanifold is contractible, using topological and differential techniques depending on curve regularity.
• Construct explicit (topologically) transversal sections to each stratum, enabling local analysis of the stratification.
• Analyze the neighboring relations between strata to understand the global structure of the stratification.
• Use the stratification to build a dual CW complex that maps weakly homotopy equivalently to the original curve space.

Experimental results

Research questions

• RQ1 Can the space of locally convex curves in $Spin_{n+1}$ be stratified in a way that reflects the sequence of Bruhat cells traversed?
• RQ2 Are the strata defined by fixed itineraries contractible and of finite codimension?
• RQ3 How do the regularity conditions of the curves affect the differentiability and transversality properties of the strata?
• RQ4 Can explicit transversal sections be constructed for each stratum to support further homotopical analysis?
• RQ5 Does this stratification support the construction of a weak homotopy equivalence to a CW complex?

Key findings

• Each stratum corresponding to a fixed itinerary word $w$ is an embedded, contractible topological submanifold of finite codimension in the space of curves.
• The stratification is not Whitney regular, indicating it lacks certain desirable smoothness properties for standard stratified Morse theory.
• Explicit topologically transversal sections exist for each stratum, enabling local study of the stratification structure.
• The construction of the stratification depends on the differentiability class of the curves, affecting the regularity of the strata and their transversality.
• The stratification provides the necessary ingredients for building a dual CW complex that is weakly homotopy equivalent to the original space of curves.
• The paper confirms a conjecture by B. Shapiro and M. Shapiro regarding the structure of such curve spaces, particularly in relation to itinerary-based stratification.