Manifolds: Geometrically Equivalent vs. Topologically Equivalent

A manifold is the general term for a geometric figure, surface, or space. Manifolds can be of any dimension, and of are great importance in mapping, and in mathematics.

The study of the manipulation of surfaces is known as topology. It is very important to understand that geometry and topology, although both dealing with geometric figures, are very different. Geometry is concerned with sizes and shapes, i.e. measuring area, radius, perimeter, volume, and so forth. However, topology, which is more relevant to this post, does not worry about specific shapes. The following definitions are very important distinctions.

Geometrically equivalent: Two objects are geometrically equivalent if they have exactly the same size, shape and dimension. Figures with this property are called congruent. Another identical statement is that if two figures can be placed on top of each other to exactly line them up, then they are congruent.

Homotopic: Two objects are homotopic if they can be continuously deformed into one another. This means that the stretching, bending or twisting of an object does not alter it topologically. Two objects that can be continuous deformed into each other in this way are called homotopic to one another. Therefore, there exists what is called an invariant in the original surface. An invariant is defined in this sense as a feature of a manifold that remains the same when it is changed is some way. The specific example of this for two homotopic manifolds is called a homotopy group.

There are actually several different homotopy groups, each of which defines features of a manifold. The simplest of these is called the fundamental group, which determines the number of holes in a surface. The mug and the torus below both have one hole in their surface; they are homotopic and equivalent topologically.




Another way to express the same idea is to consider a point on a manifold, and, starting from that point, trace any path on the given manifold, with one condition: the endpoint and the starting point of the path must coincide. When this happens, the path is called a loop. If all possible loops can be contracted into a point without leaving the given surface, than the surface has no holes and is what is called simply connected.



A demonstration that the sphere is simply connected. The loop shown, along with any other loop beginning from any other point on the sphere, can be contracted without leaving the surface.

Other surfaces, such as the torus, do not share this property.



An image of a regular torus, with three example loops drawn on its surface. It can easily be seen that some loops, such as c can be contracted to a point without leaving the surface. Others, such as a and b, cannot. Additional analysis of the set of loops on a surface yields the number of holes, known as the genus of a surface.



The triple torus is a manifold with genus 3.

The remaining homotopy groups are, simply put, higher dimensional generalizations of this. For example, the 2nd homotopy group deals with the cutting of spheres out of a surface, unlike the fundamental or first group, which deals with the cutting of circles. Therefore, there are infinitely many homotopy groups possible, each of which corresponding to a specific dimension. The "loops" for the 2nd homotopy groups will be surfaces, rather than lines, that can be contracted to a point. In addition, there is a equivalent for any dimension. For any specific manifold, the homotopy groups will assign an invariant (or a set of invariants if multiple groups are used) that identifies it as homotopic to any other manifold with an equivalent invariant.

Homeomorphic: However, the actual definition of topologically equivalent is even broader than this. Two manifolds are topologically equivalent if they are homeomorphic. Note that if two manifolds are homeomorphic, they are homotopic, but the reverse is not necessarily true.

Two manifolds are homeomorphic if there exists some function that maps each point on one manifold to a corresponding point on the other.



An example of a simple mapping (click the image to enlarge) of the number line x (red) to the number line y=x^2 (blue). Each point on x is mapped to a corresponding point on y (only the points x=1 and x=2, becoming y=1 and y=4, respectively, are shown). The above lines are actually manifolds related by the mapping y=x^2!

However, for two manifolds related by a mapping to be homeomorphic, the mapping must satisfy certain conditions, listed below. For the mapping function f that relates a set of points X to a corresponding set of points Y:

1. The function f must be continuous.
2. The function f must be one-to-one.
3. The function f must be onto.
4. The inverse of f must be continuous.

Some definitions are in order:

Continuous: Literally that each change in input causes only a small change in output. In other words, there cannot be any "jumps" or discontinuities in the function.
One-to-one: If the function f maps a point a in X to a specific point b in Y, then a is the only value in X that f maps to b.
Onto: Each value in Y corresponds to a point in X.
Inverse: The inverse of a function f is the mapping that undoes f. In other words, f maps X to Y, and the inverse of f maps Y to X.

Armed with the ideas of homotopic, homeomorphic, and mapping, one can begin to explore the world of manifolds. (see the next post)

Sources: http://en.wikipedia.org/wiki/Homotopy_groups and other various wikipedia titles, Visual Complex Analysis by Tristan Needham, The Poincare Conjecture by Donal O'Shea, http://www.regentsprep.org/Regents/math/algtrig/ATP5/OntoFunctions.htm, http://media.web.britannica.com/eb-media/58/96258-004-7747AF96.jpg