174 research outputs found
Homotopy properties of endpoint maps and a theorem of Serre in subriemannian geometry
We discuss homotopy properties of endpoint maps for affine control systems.
We prove that these maps are Hurewicz fibrations with respect to some
topology on the space of trajectories, for a certain . 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
and the horizontal-loop space with the 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
Convex pencils of real quadratic forms
We study the topology of the set X of the solutions of a system of two
quadratic inequalities in the real projective space RP^n (e.g. X is the
intersection of two real quadrics). We give explicit formulae for its Betti
numbers and for those of its double cover in the sphere S^n; we also give
similar formulae for level sets of homogeneous quadratic maps to the plane. We
discuss some applications of these results, especially in classical convexity
theory. We prove the sharp bound b(X)\leq 2n for the total Betti number of X;
we show that for odd n this bound is attained only by a singular X. In the
nondegenerate case we also prove the bound on each specific Betti number
b_k(X)\leq 2(k+2).Comment: Updated version to be published in DC
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The codimension
In this snapshot we discuss the notion of codimension, which is, in a sense, “dual” to the notion of dimension and is useful when studying the relative position of one object insider another one
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