User:Ezrapond/Notes on Differential Topology

Smooth Manifolds

Topological Manifolds

Suppose M is a topological space. We say that M is a topological manifold of dimension n or a topological n-manifold if it has the following properties:

• M is a Hausdorff space: for every pair of distinct points p, q ∈ M, there are disjoint open subsets U, V ⊆ M such that p ∈ U and q ∈ V.
• M is second-countable: there exists a countable basis for the topology of M.
• M is locally Euclidean of dimension n: each point of M has a neighborhood that is homeomorphic to an open subset of Rⁿ.

Exercise 1.1. Show that equivalent definitions of manifolds are obtained if instead of allowing U to be homeomorphic to any open set of Rⁿ, we require it to be homeomorphic to an open ball in Rⁿ, or to Rⁿ itself.

Theorem 1.2 (Topological Invariance of Dimension). A nonempty n-dimensional topological manifold cannot be homeomorphic to an m-dimensional manifold unless m=n.

Coordinate Charts

A coordinate chart(or just a chart) on M is a pair (U, φ), where U is an open subset of M and φ: U → 'Û is a homeomorphism from U to an open subset Û = φ(U) ⊆ Rn. By definition of a topological manifold, each point p ∈ M is containd in the domain of some chart.

• We call U a coordinate domain and φ a coordinate map, and the component functions (x¹, ..., xⁿ) of φ, defined by φ(p) = (x¹(p), ..., xⁿ(p)), are called local coordinates on U.

Examples

• Example 1.3 (Graphs of Continuous Functions).
  

• Example 1.4 (Spheres).
  

• Example 1.5 (Projective Spaces).
  
  • c.f. P.6.

Exercise 1.6. Show that RPⁿ is Hausdorff and second-countable, and is therefore a topological n-manifold.

Exercise 1.7. Show that RPⁿ is compact. [Hint: show that the restriction of π to Sⁿ is surjective.]

• Example 1.8 (Product Manifolds).
  
• Example 1.9 (Tori).
  

Topological Properties of Manifolds

Lemma 1.10. Every topological manifold has a countable basis of precompact coordinate balls.

Connectivity

Proposition 1.11. Let M be a topological manifold.

• M is locally path-connected.
• M is connected if and only if it is path-connected.
• The components of M are the same as its path components.
• M has countably many components, each of which is an open subset of M and a connected topological manifold.

Local Compactness and Paracompactness

Proposition 1.12 (Manifolds Are Locally Compact). Every topological manifold is locally compact.