Manifolds

Abstract

A manifold is a space which is locally like euclidean space. Some of the most important topological spaces are manifolds: Lie groups and their homogeneous spaces are manifolds. If a (compact) Lie group operates on a manifold then the orbit of every point is a manifold; if the operation is sufficiently regular then the orbit space is also a manifold. The set of solutions x = (x¹, ... , xⁿ) ∈ ℝ _n of a sufficiently regular system of equations α^μ (x¹,..., xⁿ) = 0, μ = 1,..., m, is a manifold. These and other examples justify studying the special homology properties of manifolds.