Homotopy and Homology
Let us continue the discussion about homotopy and introduce some definitions. Let π0(X) be the set of paths in X and let π0(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 “Ωp 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:
π1(X, p) = π0(Ωp X, p)
The advantage of this abstraction is that it naturally generalizes to higher levels:
πk+1(X, p) = πk(Ωp X, p)
So for example π2 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, π1 is non-abelian in general, and this points the way towards a key simplification: let’s construct an abelian (commutative) group out of π1. 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:
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.