Transverse Lusternik–Schnirelmann category of Riemannian foliations

AbstractThe transverse Lusternik–Schnirelmann category of a foliation is an invariant of foliated homotopy type. In this paper we show that the category of a Riemannian foliation is infinite if there exists a non-compact leaf verifying certain conditions. Examples of the obstruction to construct categorical coverings in such foliations are given

Morita Invariance of Equivariant Lusternik-Schnirelmann Category and Invariant Topological Complexity

We use the homotopy invariance of equivariant principal bundles to prove that the equivariant ${\mathcal A}$-category of Clapp and Puppe is invariant under Morita equivalence. As a corollary, we obtain that both the equivariant Lusternik-Schnirelmann category of a group action and the invariant topological complexity are invariant under Morita equivalence. This allows a definition of topological complexity for orbifolds

Equivariant topological complexity

We define and study an equivariant version of Farber's topological complexity for spaces with a given compact group action. This is a special case of the equivariant sectional category of an equivariant map, also defined in this paper. The relationship of these invariants with the equivariant Lusternik-Schnirelmann category is given. Several examples and computations serve to highlight the similarities and differences with the non-equivariant case. We also indicate how the equivariant topological complexity can be used to give estimates of the non-equivariant topological complexity.Comment: v1: 19 pages; v2: 14 pages. Final version, to appear in Algebraic & Geometric Topolog

Morita Invariance of Invariant Topological Complexity

We show that the invariant topological complexity defines a new numerical invariant for orbifolds. Orbifolds may be described as global quotients of spaces by compact group actions with finite isotropy groups. The same orbifold may have descriptions involving different spaces and different groups. We say that two actions are Morita equivalent if they define the same orbifold. Therefore, any notion defined for group actions should be Morita invariant to be well defined for orbifolds. We use the homotopy invariance of equivariant principal bundles to prove that the equivariant A-category of Clapp and Puppe is invariant under Morita equivalence. As a corollary, we obtain that both the equivariant Lusternik-Schnirelmann category of a group action and the invariant topological complexity are invariant under Morita equivalence. This allows a definition of topological complexity for orbifolds. This is joint work with Andres Angel, Mark Grant and John OpreaNon UBCUnreviewedAuthor affiliation: Wright CollegeFacult