## Homotopy and Homology

Let us continue the discussion about homotopy and introduce
some definitions. Let π

_{0}(X) be the set of paths in X and let π_{0}(X,p) be the set of paths containing the point p. Remembering from last time, we want to make closed loops and consider a unit circle which parameterizes a loop. Then for each point p of X we can introduce a**loop space**“Ω_{p}X” which is the space of maps from the unit circle to X and for which p is the starting point of the loop.
With those preliminaries, we can now introduce a fancy
abstract redefinition of the first homotopy group (the fundamental
group) as follows:

π

_{1}(X, p) = π_{0}(Ω_{p}X, p)
The advantage of this abstraction is that it naturally generalizes
to higher levels:

π

_{k+1}(X, p) = π_{k}(Ω_{p}X, p)
So for example π

_{2}represents the first homotopy group of the loop space meaning we cover X with a surface and we try to continuously deform it to a loop. In higher dimensions, this becomes a nightmare of visualization and we need a way to compute those groups algebraically (we need a mechanical method relatively free of visual intuition). And why do we want to compute those groups in the first place? The point is that spaces with different groups are distinct.
As a group, π

_{1 }is non-abelian in general, and this points the way towards a key simplification: let’s construct an abelian (commutative) group out of π_{1.}This will be the first homology group, but we are not yet ready to discuss it. Instead let’s try to understand what abelianization means. It means that we don’t care about the point p and we replace the closed loops with cut and stitching operations, the same way a tailor is making clothes.
For example take an inner tube which is equivalent with a
doughnut, or torus.

**If we cut the inner tube along its two circles we obtain a flat surface**. Equivalently, if we stitch (glue, zip) a rectangular piece of cloth along the opposite sides matching the stitching direction, we obtain a torus. As a nontrivial example of a torus, in early computer games, because of memory limitation, sometimes an object exits through the right edge of the screen and reappears from the left (the same behavior at top and bottom).
Probably the best way to understand cut and stitch is to try
to understand the technique of the proof of a problem mathematicians solved
around 1900: what is the complete classification of two dimensional compact
surfaces?

Two dimensional surfaces are easiest to visualize and it
turns out that orientability and Euler’s characteristic are the only ingredients
needed for a complete classification. For the record, the answer to the
classification problem is that the surface is characterized by the numbers of “holes” (or genus) and the numbers of “cross caps”.

Euler’s characteristic was discussed two posts ago, and non-orientability
is easy to understand using Mobius band.

Now it is (relatively) straightforward to solve the
classification theorem for 2 dimensional compact surfaces. All we have to do is
to cut the surface until we obtain small flat pieces, remembering the stitching
back instructions. Then we need to find some equivalent way of writing stitching
instructions in a “standard normalized way”. In the end we tally all “standard
normalized ways” and obtain the proof.

For a beautiful presentation of the proof, please see those
lectures by Norm Wildberger:

and

If you are new to algebraic topology, I highly encourage you
to invest the time and energy to watching and understanding those lectures.

The technique of the theorem’s proof will naturally introduce
us to a way of thinking about homology. Homology is related to the boundary of
a space. The link with physics is that integration on the boundary can tell us essential
things about the bulk (does Gauss theorem ring a bell?) Moreover when we jump from homology to cohomology, we’ll see
that differential forms can also reveal information about space. This is the
realm of the beautiful Hodge and de Rham theories which are essential to a
modern understanding of physics.

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