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A matchbox manifold is a generalized lamination, and is a continuum whose arc-components define the leaves of a foliation of the space. The main result of this paper implies that a matchbox manifold which is manifold-like must be…

Matchbox manifolds are foliated spaces with totally disconnected transversals. Two matchbox manifolds which are homeomorphic have return equivalent dynamics, so that invariants of return equivalence can be applied to distinguish…

We prove that a homogeneous matchbox manifold of any finite dimension is homeomorphic to a McCord solenoid, thereby proving a strong version of a conjecture of Fokkink and Oversteegen. The proof uses techniques from the theory of foliations…

A matchbox manifold is a connected, compact foliated space with totally disconnected transversals; or in other notation, a generalized lamination. It is said to be Lipschitz if there exists a metric on its transversals for which the…

We prove that (apart from dimension $n=4$), each Riemannian solenoidal lamination with transitive homeomorphism group and leaves isometric to a symmetric space $X$ of noncompact type, is homeomorphic to the inverse limit of the system of…

A matchbox manifold with one-dimensional leaves which has equicontinuous holonomy dynamics must be a homogeneous space, and so must be homeomorphic to a classical Vietoris solenoid. In this work, we consider the problem, what can be said…

A matchbox manifold is a foliated space with totally disconnected transversals, and an equicontinuous matchbox manifold is the generalization of Riemannian foliations for smooth manifolds in this context. In this paper, we develop the…

The equivariant cohomology of a space with a group action is not only a ring but also an algebra over the cohomology ring of the classifying space of the acting group. We prove that toric manifolds (i.e. compact smooth toric varieties) are…

Let M be a closed simply connected n-manifold of positive sectional curvature. We determine its homeomorphism or homotopic type if M also admits an isometric elementary p-group action of large rank. Our main results are: There exists a…

Let M be a closed simply connected 2n-dimensional manifold. The present paper is concerned with the cohomology of classifying spaces of connected groups of homeomorphisms of M.

We describe various equivalent ways of associating to an orbifold, or more generally a higher \'etale differentiable stack, a weak homotopy type. Some of these ways extend to arbitrary higher stacks on the site of smooth manifolds, and we…

We consider classification problems for manifolds and discrete subgroups of Lie groups from a descriptive set-theoretic point of view. This work is largely foundational in conception and character, recording both a framework for general…

We show that any noncompact oriented surface is homeomorphic to the leaf of a minimal foliation of a closed $3$-manifold. These foliations are (or are covered by) suspensions of continuous minimal actions of surface groups on the circle.…

Let M be a closed 5-manifold of pinched curvature 0<\delta\le \text{sec}_M\le 1. We prove that M is homeomorphic to a spherical space form if M satisfies one of the following conditions: (i) \delta =1/4 and the fundamental group is a…

In this article we consider the homotopy theory of stratified spaces through a simplicial point of view. We first consider a model category of filtered simplicial sets over some fixed poset $P$, and show that it is a simplicial…

We show that the Hamiltonian Lagrangian monodromy group, in its homological version, is trivial for any weakly exact Lagrangian submanifold of a symplectic manifold. The proof relies on a sheaf approach to Floer homology given by a relative…

Let $f\colon M\to N$ be a proper map between two aspherical compact orientable 3-manifolds with empty or toroidal boundary. We assume that $N$ is not a closed graph-manifold. Suppose that $f$ induces an epimorphism on fundamental groups. We…

In this work we relate the known results about the homotopy type of classifying spaces for smooth foliations, with the homology and cohomology of the discrete group of diffeomorphisms of a smooth compact connected oriented manifold. The…

In this paper, we study affine manifolds endowed with linear foliations. These are foliations defined by vector subspaces invariant by the linear holonomy. We show that an $n$-dimensional compact, complete, and oriented affine manifold…

Suppose M is a noncompact connected PL 2-manifold and let H(M)_0 denote the identity component of the homeomorphism group of M with the compact-open topology. In this paper we classify the homotopy type of H(M)_0 by showing that {\cal…