Lamination (topology)
In topology, a lamination is a decomposition of a space into a union of disjoint submanifolds, called leaves, that locally looks like a foliation. While a foliation is a partition of a manifold by immersed submanifolds that are locally the slices of a product structure, a lamination relaxes this requirement by allowing the "transverse structure" to be a more general compact metric space rather than a manifold.
More formally, a lamination of a compact Hausdorff space X is a closed subset L of X which is a union of disjoint, immersed submanifolds (the leaves) such that every point in L has a neighborhood U in X and a compact metric space Y, and a homeomorphism h: U -> V ⊆ Rk × Y, where k is a constant and V is an open set. Under this homeomorphism, the leaves of L intersect U in a collection of "slices" of the form Rk × {y} for y in some subset of Y. The space Y captures the transverse structure of the lamination. The integer k is called the dimension of the lamination.
A key difference between laminations and foliations is that the transverse structure of a lamination need not be locally Euclidean. This allows for more general and flexible structures.
Examples of laminations include:
- Foliations: Every foliation is a lamination.
- Cantor sets: A Cantor set in the plane can be thought of as a lamination by zero-dimensional leaves (points).
- Minimal sets of dynamical systems: The closure of a minimal orbit of a dynamical system can form a lamination.
The study of laminations is relevant in various areas of mathematics, including dynamical systems, geometric topology, and the study of group actions.