Shape of matchbox manifolds

In this work, we develop shape expansions of minimal matchbox manifolds without holonomy, in terms of branched manifolds formed from their leaves. Our approach is based on the method of coding the holonomy groups for the foliated spaces, to define leafwise regions which are transversely stable and are adapted to the foliation dynamics. Approximations are obtained by collapsing appropriately chosen neighborhoods onto these regions along a "transverse Cantor foliation". The existence of the "transverse Cantor foliation" allows us to generalize standard techniques known for Euclidean and fibered cases to arbitrary matchbox manifolds with Riemannian leaf geometry and without holonomy. The transverse Cantor foliations used here are constructed by purely intrinsic and topological means, as we do not assume that our matchbox manifolds are embedded into a smooth foliated manifold, or a smooth manifold.Comment: 36 pages. Revision of the earlier version: introduction is rewritten. Accepted to a special issue of Indagationes Mathematica

The Normal Form Theorem around Poisson Transversals

We prove a normal form theorem for Poisson structures around Poisson transversals (also called cosymplectic submanifolds), which simultaneously generalizes Weinstein's symplectic neighborhood theorem from symplectic geometry and Weinstein's splitting theorem. Our approach turns out to be essentially canonical, and as a byproduct, we obtain an equivariant version of the latter theorem.Comment: 15 pages; v2: the title was changed; v3: proof of Lemma 2 was include

Topological transversals to a family of convex sets

Let $\mathcal F$ be a family of compact convex sets in $\mathbb R^d$. We say that $\mathcal F$ has a \emph{topological $\rho$-transversal of index $(m,k)$} ($\rho<m$, $0<k\leq d-m$) if there are, homologically, as many transversal $m$-planes to $\mathcal F$ as $m$-planes containing a fixed $\rho$-plane in $\mathbb R^{m+k}$. Clearly, if $\mathcal F$ has a $\rho$-transversal plane, then $\mathcal F$ has a topological $\rho$-transversal of index $(m,k),$ for $\rho<m$ and $k\leq d-m$. The converse is not true in general. We prove that for a family $\mathcal F$ of $\rho+k+1$ compact convex sets in $\mathbb R^d$ a topological $\rho$-transversal of index $(m,k)$ implies an ordinary $\rho$-transversal. We use this result, together with the multiplication formulas for Schubert cocycles, the Lusternik-Schnirelmann category of the Grassmannian, and different versions of the colorful Helly theorem by B\'ar\'any and Lov\'asz, to obtain some geometric consequences

Lipshitz matchbox manifolds

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 holonomy maps are Lipschitz. Examples of Lipschitz matchbox manifolds include the exceptional minimal sets for $C^1$-foliations of compact manifolds, tiling spaces, the classical solenoids, and the weak solenoids of McCord and Schori, among others. We address the question: When does a Lipschitz matchbox manifold admit an embedding as a minimal set for a smooth dynamical system, or more generally for as an exceptional minimal set for a $C^1$-foliation of a smooth manifold? We gives examples which do embed, and develop criteria for showing when they do not embed, and give examples. We also discuss the classification theory for Lipschitz weak solenoids.Comment: The paper has been significantly revised, with several proofs correcte