Lerario, Antonio and Mondino, Andrea (2019) Homotopy properties of horizontal loop spaces and applications to closed sub-riemannian geodesics. Transactions of the American Math Society, Series B, 6 . pp. 187-214. doi:10.1090/btran/33 ISSN 2330-0000.

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Abstract

Given a manifold M and a proper sub-bundle Δ⊂ TM, we investigate homotopy properties of the horizontal free loop space Λ, i.e., the space of absolutely continuous maps γ: S1→M whose velocities are constrained to Δ (for example: legendrian knots in a contact manifold).

In the first part of the paper we prove that the base-point map F:Λ→M (the map associating to every loop its base-point) is a Hurewicz fibration for the W1,2 topology on Λ. Using this result we show that, even if the space Λ might have deep singularities (for example: constant loops form a singular manifold homeomorphic to M), its homotopy can be controlled nicely. In particular we prove that Λ (with theW1,2topology) has the homotopy type of a CW-complex, that its inclusion in the standard free loop space (i.e., the space of loops with no non-holonomic constraint) is a homotopy equivalence, and consequently that its homotopy groups can be computed as πk (Λ) πk (M) πk+1 (M) for all k ≥ 0.

In the second part of the paper we address the problem of the existence of closed sub-Riemannian geodesics. In the general case we prove that if (M,Δ)is a compact sub-Riemannian manifold, each non-trivial homotopy class inπ1(M) can be represented by a closed sub-Riemannian geodesic.

In the contact case, we prove a min-max result generalizing the celebrated Lyusternik-Fet theorem: if (M,Δ) is a compact, contact manifold, then every sub-Riemannian metric on Δ carries at least one closed sub-Riemannian geodesic. This result is based on a combination of the above topological results with the delicate study of an analogue of a Palais-Smale condition in the vicinity of abnormal loops (singular points of Λ).

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science, Engineering and Medicine > Science > Mathematics
Library of Congress Subject Headings (LCSH):	Geodesics (Mathematics), Loop spaces, Homotopy theory
Journal or Publication Title:	Transactions of the American Math Society, Series B
Publisher:	American Mathematical Society
ISSN:	2330-0000
Official Date:	6 May 2019
Dates:	Date Event 6 May 2019 Published 12 November 2018 Accepted
Volume:	6
Page Range:	pp. 187-214
DOI:	10.1090/btran/33
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Open Access (Creative Commons)
Date of first compliant deposit:	5 December 2018
Date of first compliant Open Access:	7 May 2019
RIOXX Funder/Project Grant:	Project/Grant ID RIOXX Funder Name Funder ID UNSPECIFIED Eidgenössische Technische Hochschule Zürich http://dx.doi.org/10.13039/501100003006 UNSPECIFIED [SNSF] Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung http://dx.doi.org/10.13039/501100001711 EP/R004730/1 [EPSRC] Engineering and Physical Sciences Research Council http://dx.doi.org/10.13039/501100000266
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