We study the topology of the space of admissible paths between two points
(the origin) and
on a step-two Carnot group
:
admissible,
.
As it turns out, is homotopy equivalent to an infinite dimensional sphere and in particular it is contractible. The energy function:
is defined by ;critical points of this function are sub-Riemannian geodesics between
and
. We study the asymptotic of the number of geodesics and the topology of the sublevel sets:
as
.
If p is not a vertical point in , the number of geodesics joining
and
is bounded and the homology of
stabilizes to zero for
large enough.
A completely different behavior is experienced for the generic vertical . In this case we show that
is a Morse-Bott function: geodesics appear in isolated families (critical manifolds), indexed by their energy. Denoting by
the corank of the horizontal distribution on
, we prove that:
Card{Critical manifolds with energy less than }
.
Despite this evidence, Morse-Bott inequalities are far from being sharp and we show that the following stronger estimate holds:
. Thus each single Betti number
becomes eventually zero as
, but the sum of all of them can possibly increase as fast as
. In the case
we show that indeed
The leading order coefficient
can be analytically computed using the structure constants of the Lie algebra of
.
Using a dilation procedure, reminiscent to the rescaling for Gromov-Hausdorff limits, we interpret these results as giving some local information on the geometry of (e.g. we derive for
the rate of growth of the number of geodesics with bounded energy as
approaches
along a vertical direction).