A Quick Intro to Manifolds

An n-dimensional manifold is intuitively described as a topological space that locally resembles \mathbb{R}^n.  In other words, if you pick a point on a manifold and “zoom in”, it starts to look more and more like a Euclidean space of the same dimension. According to Penrose a manifold is

… a space that can be thought of as ‘curved’ in various ways, but where, locally (i.e. in a small enough neighborhood of any of its points), it looks like a piece of ordinary Euclidean space.

Some examples will help to illustrate.

Example: If we start with n = 0, then \mathbb{R}^{0} = \{0\}, which is just a singleton.  This means that a 0-manifold is one in which every point has a neighborhood that looks like a single point.  Thus, a 0-manifold just is a collection of points.

Example: If n=1, then \mathbb{R}^{1} is the real line.  Thus, a 1-manifold is one in which each point has a neighborhood that looks like an open line segment.  A circle is a good example of this.  In topology, one ignores “bending” so that a small portion of the circle is treated like a small line segment.  Let’s consider this more carefully by considering the unit circle x^{2}+y^{2} = 1 and seeing how it “looks like” a line segment at each point.

LUFTOCirc

Begin with the upper half part of the circle described by y = \sqrt{1-x^{2}},~ -1<x<1.

LUFTOUpperHalfCirc

Note that the y-coordinate is always positive on this part of the circle.  Since any point on this portion of the circle can be uniquely described by its x-coordinate we can use the projection map \rho_{top}: S^{(+)}\to (-1,1) defined by

\rho_{top}(x,y) = x

to map the open upper half-circle S^{(+)} to the open interval (-1,1).

This mapping is both continuous and invertible.  Some example projections are shown in the diagram below.

LUFTOUpperHalfProj

The pair (\rho_{top}, S^{(+)}) is called a chart.  There are similar charts for the open lower half-circle, the open right half-circle and the open left half-circle.  Taken together, these charts cover the whole circle forming an atlas. 

Example:  For n=2, the space \mathbb{R}^{2} is the plane.  Thus, a 2-manifold is one in which every point has a neighborhood that looks like an open portion of the plane.  The surface of a sphere or a torus are good examples.

The formal definition of a manifold is a bit involved, so let’s start with the condensed version and unpack it from there.

Definition[Manifold]: A topological manifold is a second-countable Hausdorff  space that is locally homeomorphic to Euclidean space.

Let’s first look at what it means for a space to be “second-countable”.

Definition[Second-Countable]: A space is said to be second-countable if its topology has a countable base.

Here a base B for a topological space X with topology T is a collection of open sets in T such that every open set in T can be written as a union of elements from B.  The base B is said to generate the topology T.

That B is countable simply means that there exists an injective function f: B\to \mathbb{N}.  In other words, the elements of B can be put in one-to-one correspondence with a subset of the natural numbers.

Next, let’s consider Hausdorff spaces.

Definition[Hausdorff Space]:  A Hausdorff space H is a topological space such that for all x,y\in H with x\neq y, there exist neighborhoods U, V in the topology with x\in U, y\in V, and U\cap V = \varnothing.

Sometimes Hausdorff spaces are referred to as separated spaces.  This is illustrated below.

LUFTOHausdorff

As Penrose puts it

A Hausdorff space has the defining property that, for any two distinct points of the space, there are open sets containing each which do not intersect.

The last bit we need is to understand what it means to be locally homeomorphic to Euclidean space.

Definition[Homeomorphism]:  Let S and T be two topological spaces.  A homeomorphism h is a function h:S\to T that satisfies the following properties:

  1. h is a bijection.
  2. h is continuous.
  3. h^{-1} is continuous.

Definition[Local Homeomorphism]:  Let S and T be two topological spaces.  A function f:S\to T is called a local homeomorphism if for every point x\in S there exists an open set U containing x, such that the image f(U) is open in T, and the restriction f|_{U}: U\to f(U) is a homeomorphism.

So, to be locally homeomorphic to Euclidean space means that every point of an n-manifold has a neighborhood that is homeomorphic to an open Euclidean n-ball

B^{n} = \{(x_{1}, x_{2}, ..., x_{n}) \in \mathbb{R}^{n}: x_{1}^{2}+x_{2}^{2}+\ldots +x_{n}^{2}<1\}

In other words, if H is a second-countable Hausdorff space, then it is locally homeomorphic to Euclidean space iff there exists a function f: H\to \mathbb{R}^{n} such that for each x\in H there is a neighborhood U of x such that f|_{U}:U\to B^{n} is a homeomorphism (where f(U) = B^{n}).

References:

Differential Forms: A Complement to Vector Calculus, by Steven H. Weintraub

The Road to Reality, by Roger Penrose

Fundamentals of Topology, by Benjamin T. Sims

http://en.wikipedia.org/wiki/Second_countable

http://en.wikipedia.org/wiki/Manifold

http://en.wikipedia.org/wiki/Base_(topology)

http://en.wikipedia.org/wiki/Hausdorff_space