Computational Algebraic Topology, lecture notes [pdf]
people.maths.ox.ac.uk
people.maths.ox.ac.uk
These notes contain the definition of various chain complexes and groups (co-homology/homotopy) that are in-principle computable. However, I have not seen this being efficiently applied somewhere, and not been able to come up with interesting applications myself. Topics I have considered are:
1. Triangulation of solutions of equations (for e.g. homology computations) does not seem to be very popular, at least in dimension >3. I suspect this is a hard problem, but maybe I am just missing pointers to the right literature.
2. CAD applications or Computer Games have lot's of triangulated objects. Their topology seems to be not to be very interesting. Again: In dimension <=3 there is really not that much going on. And since you have constructed the object yourself, you probably know the geometry already.
3. Graphs appear everywhere, and can be viewed as 1-dimensional chain complexes but do not have interesting co-homology groups.
4. Did anyone compute homology groups for "Manifold Learning"?
It's also not clear to me, how much interesting information can be extracted from those homology groups. Applications of Homotopy/Homology in (semi-classical) Physics are already quite slim (apart from Quantum Field Theory, String Theory, Gauge Theory?!) as most of it takes place in contractible spaces $IR^n$.
The one practical algorithm I've seen which uses topological arguments for justification is UMAP for dimensionality reduction, the core idea IMO* is very much like density estimation and I'm not sure the mathematical justification gives more insight.
* The core idea IMO: while in high-dimensional space, absolute distances lose meaning, relative distances between neighboring points are good yard-sticks. This insight is also what powers DBSCAN and other such density-based algorithms. I don't think using the language of simplical complexes when explaining the algorithm simplifies or adds insight.
https://link.springer.com/book/10.1007/978-3-642-36104-3
You can download it for free from academic libraries that have SpringerLink.
If you're coming from topology to AT and can grok the ker/im stuff, TDA should be easy peasy to you. I flunked (I think I scraped a couple of points on appeal by sophistry and desperation so I passed, but really) real analysis, so I never moved into topology beyond skimming books to "learn about" key concepts.
You simply can't have too much math in life. I'm having a kid soon and I struggle between what seems truer to my core values (let the kid have a childhood and then personal inclinations; advise but not push; if he wants to do the gifted kid thing, yay) and what seems practical advice (learn as many human languages as you can, at least English, French and German; learn math deeply, suffering along the way).
To be even more concrete, if I cut out a disk from a piece of paper, I've created a cycle (the circle bounding the disk) that is not the boundary of any disk in the piece of paper (because I cut it out).
Admittedly, this gets a little funkier in higher dimensions, but for 2 dimensions, it's a good guide. (I think Hatcher's book has a discussion of how to make this idea precise in higher dimensions.)
And on the parenting front, luckily you start with a baby not a ten year old so you have lots of chance to grow as a parent before making those sorts of decisions - a lot of parenting is deciding when to be directive/non-directive and when to take charge based on your superior understanding of the world (not letting your toddler walk into the road!) and when to let them learn from their own mistakes.
He kindly provides a pdf on his webpage here: https://pi.math.cornell.edu/~hatcher/AT/ATpage.html
What's a cycle? It's something in the kernel of the boundary map. To be in the kernel means the boundary map takes it to 0. So heuristically, cycles are "things with no (zero) boundary".
For the quotient cycles/boundaries to make sense - for any quotient to make sense - the bottom should be contained in the top. Like in Z/6Z, the integers mod 6, the "6Z" (the integers which are a multiple of 6) is contained in Z (the integers).
In this case, for "cycles/boundaries" to make sense, a boundary should be a cycle - that is, a boundary should have no boundary. This is (again heuristically) what the definition of a boundary map (d^2 = 0) means.
Consider your disk/circle analogy. The circle is a boundary (the boundary of the disk). Does the circle have a "boundary"? No - the boundary of a boundary is 0.
When the quotient cycles/boundaries is nonzero, you have nontrivial homology. Think of nontrivial homology as "having a hole". To see an example of this, take the plane but remove the origin (0, 0). Consider your circle again (say the unit circle x^2 + y^2 = 1). It's still a cycle, because its boundary is 0 (no boundary). But it's not the boundary of "something" (where "something" means roughly "something nice" - it's hard to explain this better without getting technical). So the cycle represented by the circle represents a nontrivial homology class (in the homology of plane minus origin).
I'm going to be hand wavey here, but in DeRham, the "boundary" operator is basically just the derivative. So the DeRham homology of a manifold (the ker/im) are the functions that differentiate to zero (locally constant functions) / functions that can be integrated (eg functions that are the derivative of some function). You get left the functions that are "interesting" and define the geometry of the manifold in some way. Since manifolds are patched together pieces of Euclidean space, locally constant functions don't need to be globally constant which is why the quotient operation isn't trivially zero. If you've never seen quotient rings/groups/topologies before though, I can see how ker/im would be a major trip-up no matter the homology domain.
The crazy part of algebraic topology (to me) is that all these homology theories are isomorphic. The DeRham homology and the simplicial homology give you the same homology group despite not having much to do with each other at first glance. (Stoke's theorem provides the isomorphism).
But I kind of have a BDSM relationship with mathematics. I'm not happy until it makes me suffer.
The standard cohomology groups with integer coefficients H^n(X; Z), those are just homotopy classes of maps from X into the Eilenberg-MacLane K(Z,n). Want K-theory instead? No problem. That cohomology group K(X) is homotopy classes of maps [X, Z x BU], where Z is the integers and BU is the classifying space for the unitary group.
It turns out essentially all cohomology theories are represented this way. h(X) = [X,S_h], where [] is homotopy classes of maps and S_h is a topological space called the spectrum of the cohomology theory.
Even wackier, the various algebra operations you can do on cohomology classes are mirrored by / derived from algebraic operations on the spectra. Which means that much of the machinery of algebraic geometry can be used to study cohomology theories.
This should not be crazy at all. The general recipe is that you take something for which there is a local solution, say in a ball or square, and then ask what are the obstructions that prevent you from patching the local solutions into a global solution. That is all that's going on with the ker/Im.
For example, differential equations have local solutions (just the contraction mapping theorem). But can you patch them into global solutions? Locally spaces can be described as a ball or disk, but can you globally patch them together? So the obstructions end up measuring something topological. A series of local solutions on, say, a circle can patch together smoothly but when you wrap around to where you started you can be off - there is an obstruction.
So the technique of finding local solutions can involve solving a differential equation or constructing a simplex but the obstructions of both have only to do with the consistency of patching these together, which is a topological question.
As a historical note, mathematicians understood that counting the dimension of these obstructions reveals something about the underlying space, and there was hope that they would end up measuring finer structures -- e.g. the differential structure rather than just the topological structure. But the above discussion should at least motivate that this is unlikely. All the various *-homology theories ended up being mostly equivalent. It wasn't until the 1980s with the discovery of Simon Donaldson that true differential invariants were found, and the mechanism of finding these was to take the old homology invariants but apply them to an associated object whose own topology is shaped by the differential structure of the original space. For example, the moduli space of vector bundles associated to a space. Then the homological invariants of that space can detect differential invariants of the original space. There is a beautiful lecture series here: https://cmsa.fas.harvard.edu/donaldson-thomas-gromov-witten/
Anyway, where is this Topological Data Analysis that's going to be easy for me?
https://www2.math.upenn.edu/~ghrist/notes.html
(I haven't gone through either.)
> Definition (Homomorphism) There is a homomorphism (i.e. an equivalence relation) between two topological spaces if there exists a function f:X→Y, where X and Y are topological spaces (X,τX) and (Y,τY) with the following properties
This should be "homeomorphism", not "homomorphism". A homomorphism is a structure-preserving map, like a continuous function (in the context of topological spaces) or a linear map (in the context of vector spaces). Homomorphisms are very much one-directional. Homeomorphisms, on the other hand, are specifically equivalences between topological spaces, and are defined as continuous functions with continuous inverses.
Edit: finished reading it. I'm impressed by how much you cover in such a short space. Overall it looks good, so thanks for taking the time to write it. Some of these ideas are at a pretty high level of abstractness, so I sympathize with anyone who struggles to get any kind of intuition for them at a first go.