Mathematics.

differential topology

Manifolds

Topology45 minDifficulty8 out of 10

You should know: hausdorff spaces, homeomorphism

Overview

A topological n-manifold is a topological space that locally 'looks like' ordinary Euclidean space ℝⁿ, even if its global shape is very different. Formally, it is a Hausdorff, second-countable topological space in which every point has an open neighborhood homeomorphic to an open subset of ℝⁿ. This local-Euclidean condition lets manifolds carry local coordinate systems (charts), while the global topology can be as rich as a sphere, torus, or higher-dimensional analogue. Manifolds are the natural setting for differential geometry and physics — a circle is a 1-manifold, a sphere or torus is a 2-manifold, and the spacetime of general relativity is modeled as a 4-dimensional manifold — because locally Euclidean structure is exactly what is needed to do calculus while still allowing the global topology to be exotic.

Intuition

Think of the surface of the Earth: zoomed in on any small patch, it looks flat, just like a piece of ordinary 2-dimensional Euclidean plane, which is why local maps (atlases) work perfectly well. But zoom back out, and the whole surface is a sphere, which is a fundamentally different global shape from a flat plane — you cannot flatten the whole sphere onto a plane without distortion or tearing. A manifold captures exactly this idea: locally indistinguishable from flat Euclidean space (so calculus, coordinates, and local geometry all make sense), while globally it may have curvature, holes, or a completely different overall topology. The name 'manifold' reflects that many local Euclidean 'folds' (charts) are stitched together to cover the whole space.

Formal Definition

Definition

A topological space X is an n-dimensional (topological) manifold if it satisfies:

∀x∈X, ∃ open U∋x and a homeomorphism φ:U→V⊆Rn (V open)\forall x \in X,\ \exists \text{ open } U \ni x \text{ and a homeomorphism } \varphi: U \to V \subseteq \mathbb{R}^n \text{ (}V\text{ open)}

Every point has a neighborhood homeomorphic to an open subset of ℝⁿ; the pair (U, φ) is called a chart

Locally Euclidean
X is HausdorffX \text{ is Hausdorff}

Needed to rule out pathologies like the 'line with two origins,' which is locally Euclidean but not Hausdorff

Hausdorff condition
X is second-countable (has a countable basis for its topology)X \text{ is second-countable (has a countable basis for its topology)}

A technical condition ruling out spaces glued from uncountably many disjoint Euclidean pieces; needed for partitions of unity and embedding theorems

Second countability
{(Uα,φα)}α covering X is called an atlas\{(U_\alpha, \varphi_\alpha)\}_{\alpha} \text{ covering } X \text{ is called an atlas}

A collection of charts whose domains cover X, used to give X coordinates locally everywhere

Atlas

Notation

NotationMeaning
(U,φ)(U,\varphi)A chart: an open set U with a homeomorphism φ to an open subset of ℝⁿ
{(Uα,φα)}\{(U_\alpha,\varphi_\alpha)\}A collection of charts covering the whole manifold

Properties

Circle S¹ is a 1-manifold

Every point of S1 has a neighborhood homeomorphic to an open interval in R.\text{Every point of } S^1 \text{ has a neighborhood homeomorphic to an open interval in } \mathbb{R}.

Sphere Sⁿ and torus Tⁿ are n-manifolds

S2,T2 are 2-manifolds; more generally Sn and Tn=(S1)n are n-manifolds.S^2, T^2 \text{ are 2-manifolds; more generally } S^n \text{ and } T^n = (S^1)^n \text{ are } n\text{-manifolds}.

Line with two origins is not Hausdorff

A space glued from two copies of R sharing all points except the origin is locally Euclidean but fails Hausdorff, hence is excluded from being a manifold.\text{A space glued from two copies of } \mathbb{R} \text{ sharing all points except the origin is locally Euclidean but fails Hausdorff, hence is excluded from being a manifold.}

Manifolds with boundary

A manifold with boundary allows charts into half-spaces {x∈Rn:xn≥0}, e.g. the closed disk D2.\text{A manifold with boundary allows charts into half-spaces } \{x\in\mathbb{R}^n : x_n \geq 0\}\text{, e.g. the closed disk } D^2.

Worked Examples

  1. Cover S¹ with four open arcs, e.g. where x>0, x<0, y>0, y<0.

    U1={(x,y)∈S1:x>0}U_1 = \{(x,y)\in S^1 : x>0\}

  2. On U₁, the map φ₁(x,y) = y is a homeomorphism onto the open interval (-1,1) ⊆ ℝ (with inverse y ↦ (√(1-y²), y)).

    φ1(x,y)=y:U1→(−1,1)\varphi_1(x,y) = y : U_1 \to (-1,1)

  3. Similar charts cover the other three arcs, and together the four charts cover all of S¹.

    {(Ui,φi)}i=14 is an atlas for S1\{(U_i, \varphi_i)\}_{i=1}^4 \text{ is an atlas for } S^1

Answer: S¹ is a 1-manifold: every point has a neighborhood (one of the four arcs) homeomorphic to an open interval in ℝ.

Practice Problems

Difficulty 5/10

A topological n-manifold is defined as a space that is:

Difficulty 5/10

What dimension manifold is the torus T², and what is a chart around a typical point?

Difficulty 8/10

Prove that a compact manifold cannot be homeomorphic to ℝⁿ (for n ≥ 1).

Quiz

A topological n-manifold requires every point to have a neighborhood that is:
Why is the Hausdorff condition included in the definition of a manifold, rather than following automatically from being locally Euclidean?
Which of these is an example of a 2-dimensional manifold?

Summary

  • A topological n-manifold is a Hausdorff, second-countable space in which every point has a neighborhood homeomorphic to an open subset of ℝⁿ.
  • Charts give local coordinates; an atlas is a collection of charts covering the whole manifold, enabling calculus to be done locally even on globally curved or exotic spaces.
  • Manifolds range from simple examples like the circle (1-manifold) and sphere/torus (2-manifolds) to the 4-dimensional spacetime manifold of general relativity, and the Hausdorff/second-countable conditions rule out pathological locally-Euclidean spaces like the line with two origins.

References

  1. WebsiteWikipedia — Manifold