## Publication Date:

6 May 2019

## Journal:

Transactions of the American Mathematical Society

## Last Updated:

2021-10-08T11:53:48.533+01:00

## DOI:

10.1090/btran/33

## abstract:

Given a manifold $M$ and a proper sub-bundle $\Delta\subset TM$, we study

homotopy properties of the horizontal base-point free loop space $\Lambda$,

i.e. the space of absolutely continuous maps $\gamma:S^1\to M$ whose velocities

are constrained to $\Delta$ (for example: legendrian knots in a contact

manifold).

A key technical ingredient for our study is the proof that the base-point map

$F:\Lambda \to M$ (the map associating to every loop its base-point) is a

Hurewicz fibration for the $W^{1,2}$ topology on $\Lambda$. Using this result

we show that, even if the space $\Lambda$ 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 $\Lambda$ (with

the $W^{1,2}$ topology) has the homotopy type of a CW-complex, that its

inclusion in the standard base-point free loop space (i.e. the space of loops

with no non-holonomic constraint) is a homotopy equivalence, and consequently

its homotopy groups can be computed as $\pi_k(\Lambda)\simeq \pi_k(M) \ltimes

\pi_{k+1}(M)$ for all $k\geq 0.$

These topological results are applied, in the second part of the paper, to

the problem of the existence of closed sub-riemannian geodesics. In the general

case we prove that if $(M, \Delta)$ is a compact sub-riemannian manifold, each

non trivial homotopy class in $\pi_1(M)$ 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 $(M, \Delta)$ is a compact, contact manifold, then

every sub-riemannian metric on $\Delta$ carries at least one closed

sub-riemannian geodesic. This result is based on a combination of the above

topological results with a delicate study of the Palais-Smale condition in the

vicinity of abnormal loops (singular points of $\Lambda$).

homotopy properties of the horizontal base-point free loop space $\Lambda$,

i.e. the space of absolutely continuous maps $\gamma:S^1\to M$ whose velocities

are constrained to $\Delta$ (for example: legendrian knots in a contact

manifold).

A key technical ingredient for our study is the proof that the base-point map

$F:\Lambda \to M$ (the map associating to every loop its base-point) is a

Hurewicz fibration for the $W^{1,2}$ topology on $\Lambda$. Using this result

we show that, even if the space $\Lambda$ 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 $\Lambda$ (with

the $W^{1,2}$ topology) has the homotopy type of a CW-complex, that its

inclusion in the standard base-point free loop space (i.e. the space of loops

with no non-holonomic constraint) is a homotopy equivalence, and consequently

its homotopy groups can be computed as $\pi_k(\Lambda)\simeq \pi_k(M) \ltimes

\pi_{k+1}(M)$ for all $k\geq 0.$

These topological results are applied, in the second part of the paper, to

the problem of the existence of closed sub-riemannian geodesics. In the general

case we prove that if $(M, \Delta)$ is a compact sub-riemannian manifold, each

non trivial homotopy class in $\pi_1(M)$ 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 $(M, \Delta)$ is a compact, contact manifold, then

every sub-riemannian metric on $\Delta$ carries at least one closed

sub-riemannian geodesic. This result is based on a combination of the above

topological results with a delicate study of the Palais-Smale condition in the

vicinity of abnormal loops (singular points of $\Lambda$).

## Symplectic id:

1061626

## Download URL:

## Submitted to ORA:

Submitted

## Publication Type:

Journal Article