[Paper Review] Homotopy properties of endpoint maps and a theorem of Serre in subriemannian geometry

Francesco Boarotto, Antonio Lerario|arXiv (Cornell University)|Feb 26, 2015

Homotopy and Cohomology in Algebraic Topology7 references3 citations

TL;DR

This paper establishes that the endpoint map in affine control systems is a Hurewicz fibration under the $W^{1,p}$ topology for $1 < p < p_c$, with $p_c = ty$ in the subriemannian case ($X_0 = 0$). It proves that on a compact base manifold, the number of critical points of geometric costs is infinite, extending Serre’s theorem on infinitely many geodesics to subriemannian geometry, and shows the horizontal loop space with $W^{1,2}$ topology has the homotopy type of a CW-complex.

ABSTRACT

We discuss homotopy properties of endpoint maps for affine control systems. We prove that these maps are Hurewicz fibrations with respect to some $W^{1,p}$ topology on the space of trajectories, for a certain $p&gt;1$. We study critical points of geometric costs for these affine control systems, proving that if the base manifold is compact then the number of their critical points is infinite (we use Lusternik-Schnirelmann category combined with the Hurewicz property). In the special case where the control system is subriemannian this result can be read as the corresponding version of Serre's theorem, on the existence of infinitely many geodesics between two points on a compact riemannian manifold. In the subriemannian case we show that the Hurewicz property holds for all $p\geq1$ and the horizontal-loop space with the $W^{1,2}$ topology has the homotopy type of a CW-complex (as long as the endpoint map has at least one regular value); in particular the inclusion of the horizontal-loop space in the ordinary one is a homotopy equivalence.

Motivation & Objective

• To establish homotopy-theoretic properties of the endpoint map in affine control systems under $W^{1,p}$ topology.
• To extend Serre’s classical result on infinitely many geodesics on compact Riemannian manifolds to the subriemannian setting.
• To analyze the topology of horizontal path spaces and loop spaces using Lusternik-Schnirelmann category and Hurewicz fibrations.
• To show that the inclusion of the horizontal loop space into the full loop space is a homotopy equivalence under $W^{1,2}$ topology in the subriemannian case.
• To characterize the regularity threshold $p_c$ for the Hurewicz fibration property depending on the control system structure.

Proposed method

• Use of $W^{1,p}$ topology on the space of controls $u = (u_1, ..., u_d) \in L^p([0,1], \mathbb{R}^d)$ to define a Banach manifold structure on the space of horizontal trajectories.
• Prove the endpoint map $F: \Omega \to M$, $\gamma \mapsto \gamma(1)$, is a Hurewicz fibration for $1 \leq p < p_c$, using the homotopy lifting property.
• Apply the implicit function theorem and linearization of the endpoint map via the variational equation to show $C^1$-differentiability of $F$.
• Use the $L^p$-norm of controls to define geometric costs and analyze their critical points via Lusternik-Schnirelmann category.
• Establish uniform convergence of the monodromy matrix $N_{u_n} \to N_u$ and continuity of the differential $d_uF$ under weak $L^p$ convergence.
• Leverage smoothness of vector fields and compactness of trajectories to derive uniform estimates on the second variation and perturbations of the flow.

Experimental results

Research questions

• RQ1 Under what topologies on the space of trajectories is the endpoint map a Hurewicz fibration?
• RQ2 What is the critical regularity threshold $p_c$ for the Hurewicz fibration property in affine control systems?
• RQ3 Does the subriemannian case ($X_0 = 0$) admit the Hurewicz fibration property for all $p \geq 1$?
• RQ4 Can the Lusternik-Schnirelmann category be used to prove the existence of infinitely many critical points of geometric costs on compact manifolds?
• RQ5 Is the inclusion of the horizontal loop space into the full loop space a homotopy equivalence under the $W^{1,2}$ topology in the subriemannian case?

Key findings

• The endpoint map $F: \Omega \to M$ is a Hurewicz fibration for the $W^{1,p}$ topology whenever $1 \leq p < p_c$, where $p_c > 1$ depends on the control system $\mathcal{F}$.
• In the subriemannian case ($X_0 = 0$), the endpoint map is a Hurewicz fibration for all $1 \leq p < \infty$, i.e., $p_c = \infty$.
• For compact base manifolds, the number of critical points of geometric costs is infinite, as shown by combining the Hurewicz fibration property with the Lusternik-Schnirelmann category.
• The horizontal loop space with the $W^{1,2}$ topology has the homotopy type of a CW-complex, provided the endpoint map has at least one regular value.
• The inclusion of the horizontal loop space into the full loop space is a homotopy equivalence under the $W^{1,2}$ topology in the subriemannian case.
• The differential $d_uF$ of the endpoint map is continuous under weak $L^p$ convergence of controls, and $F$ is $C^1$-differentiable on $W^{1,p}$ for $p > 1$.