We discuss homotopy properties of endpoint maps for affine control systems. We prove that these maps are Hurewicz fibrations with respect to some W1,p topology on the space of trajectories, for a certain p>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≥1 and the horizontal-loop space with the W1,2 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.

Homotopy properties of horizontal path spaces and a theorem of Serre in subriemannian geometry / Boarotto, Francesco; Lerario, Antonio. - In: COMMUNICATIONS IN ANALYSIS AND GEOMETRY. - ISSN 1019-8385. - 25:2(2017), pp. 269-301. [10.4310/CAG.2017.v25.n2.a1]

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

Boarotto, Francesco;Lerario, Antonio
2017-01-01

Abstract

We discuss homotopy properties of endpoint maps for affine control systems. We prove that these maps are Hurewicz fibrations with respect to some W1,p topology on the space of trajectories, for a certain p>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≥1 and the horizontal-loop space with the W1,2 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.
2017
25
2
269
301
http://dx.doi.org/10.4310/CAG.2017.v25.n2.a1
https://arxiv.org/abs/1502.07452
http://people.sissa.it/~lerario/Antonio_Lerario/Papers_files/serre_submit2.pdf
Boarotto, Francesco; Lerario, Antonio
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.11767/32844
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