Enrique Macías-Virgós

Transverse Lusternik-Schnirelmann category of foliated manifolds

Abstract The purpose of this paper is to develop a transverse notion of Lusternik–Schnirelmann category in the field of foliations. Our transverse category, denoted by cat ⋔ (M, F ) , is an invariant of the foliated homotopy type which is finite on compact manifolds. It coincides with the classical notion when the foliation is by points. We prove that for any foliated manifold cat M⩽ cat L cat ⋔ (M, F ) , where L is a leaf of maximal category, thus generalizing a result of Varadarajan for fibrations. Also we prove that cat ⋔ (M, F ) is bounded below by the index of k ∗ H b + (M) , the latter being the image in HDR(M) of the algebra of basic cohomology in positive degrees. In the second part of the paper we prove that cat ⋔ (M, F ) is a lower bound for the number of critical leaves of any basic function provided that F is a foliation satisfying certain conditions of Palais–Smale type. As a consequence, we prove that the result is true for compact Hausdorff foliations and for foliations of codimension one. This generalizes the classical result of Lusternik and Schnirelmann about the number of critical points of a smooth function.

Manifolds of Maps in Riemannian Foliations

Let (M′,F′) and (M,F) be two (compact or not) foliated manifolds, C∞F(M′, M) the space of smooth maps which send leaves into leaves. In this paper we prove that C∞F(M′, M) admits a structure of an infinite-dimensional manifold modeled on LF-spaces, provided that F is a Riemannian foliation or, more generally, when it admits an adapted local addition.

Minimal foliations on lie groups

Abstract Let F(G,H) be the foliation determined by a (not necessarily closed) Lie subgroup H of a Lie group G. We prove that there always exists a Riemannian metric on G for which the leaves of F(G,H) are minimal submanifolds.

Morse Theory and the Lusternik–Schnirelmann Category of Quaternionic Grassmannians

The Lusternik-Schnirelmann category, catX, of a topological space X is defined as the least integer m ≥ 0 such that X admits a covering by m + 1 open sets which are contractible in X. This homotopy invariant is often difficult to compute, specially in the context of spaces of quaternionic matrices. For instance, in the case of the symplectic groups, Sp(n), we only know some low values, as cat Sp(2) = 3 [18], cat Sp(3) = 5 [4], and some bounds as cat Sp(n) ≥ n + 2 when n ≥ 3 [8] or cat Sp(n) ≤ ( n+1 2 ) [10]. In the case of quaternionic Stiefel manifolds, there is also a partial result, catXn,k = k when n ≥ 2k [14]. Still, we know the LS-category of the quaternionic Grassmann manifolds Gn,k = Sp(n)/(Sp(k) × Sp(n − k)). First, recall the existence of lower and upper bounds for the LS-category. By definition, the cup length of a space X, cupX, is the largest integer ` such that there exists a product x1 · · ·x` 6= 0, with xi ∈ H̃∗(X;A), for any coefficient ring A. Then, if X is an (n− 1)-connected CW -complex, we have (see [2])