Homotopy properties of horizontal path spaces and a theorem of Serre in subriemannian geometry

We discuss homotopy properties of endpoint maps for horizontal path spaces, i.e. spaces of curves on a manifold $M$ whose velocities are constrained to a subbundle $\Delta \subset TM$ in a nonholonomic way. We prove that for every $1 \leq p \lt \infty$ these maps are Hurewicz fibrations with respect to the $W^{1,p}$ topology on the space of trajectories.

We prove that the space of horizontal curves joining any two points (with the induced $W^{1,p}$ topology) has the homotopy type of a CW-complex and its inclusion into the standard path space (i.e. with no nonholonomic constraints) is a homotopy equivalence. We derive topological implications on the local structure of these spaces (even near abnormal curves, whose possible existence is not excluded from our constructions).

We consider indeed the more general class of affine control systems, for which the above theorems hold for all $1 \leq p \lt p_c$ (here $p_c \gt 1$ depends only the step of the system).

We study critical points of geometric costs for these affine control systems, proving that if the base manifold is compact and there are no abnormal trajectories, 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.

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Received 26 February 2015

Published 4 August 2017