Applications of algebraic topology?

Question

Terribly sorry if this has been asked, but I'm not about to search 382 pages of technical questions in the field.

I am trying to develop a very basic understanding of what algebraic topology is about. Tried Hatcher, who simply dives into vague terminology I've not heard for reasons that are not clear. The heck is an $n$-cell...? What do you mean, "attached?" How in the heck are $f'$ and $f''$ from $X\to Y$ "connected" by a homotopy...?

I go into May's "Concise" book, which has at least clarified the fundamental idea of the field (algebraic invariants under continuous deformations of the topology, I gather), but the first two sentences leave me wondering how intuitive this book is actually going to be, when it comes to structures with which I'm not already familiar. And he clarifies what Hatcher meant by "connected": $h(x,0) = p(x)$ and $h(x,1) = q(x)$ "connects" $p$ and $q$ by a homotopy $h$... I can see that. But extrapolating from many answers on this site I have read, I will be judged harshly for not simply intuiting this from the vague wording. That's okay, you can think I'm stupid. I've got no problem with that.

But what I'm grasping for, and not finding anywhere, is why I should be interested in algebraic topology in the first place. What does it actually tell us about geometry? I am interested in what can be done in a space where algebraic structures is commensurable with continuous maps, but I am seriously questioning whether the field, as it is, has any true insight to offer me here. It doesn't seem like they're investigating structures with any real relevance to anything, anywhere. I am sure that I am simply lacking imagination here. I'm struggling enough to even grasp the basic concepts in any other light than pure symbol-games, let alone understand what the 'big' theorems are about. Why should I care about fundamental groups or homotopy type? What does it MEAN...? What is algebraic topology REALLY about...? What's the point of it? What does it apply to? I am sure it applies to many results in higher math, but this doesn't interest me. What disciplines in sciences and engineering have made the greatest use of results and language of algebraic topology?

I am starting to have the feeling that nobody really has any idea what they're talking about in these fields. Please prove me wrong. I am utterly perplexed as to what the point is.

Answer 1

The power of algebraic topology is that problems which seem to have little or nothing to do with algebra, or little or nothing to do with topology, can be converted into algebraic topology questions, and from there into purely algebraic computations, and thereby can be solved.

Rather than trying to convince you on some philosophical level, let me list some mathematical facts which can be proved by applying algebraic topology. I am choosing these applications to be as broad and as unconnected with each other as I can imagine. One of these has a non algebraic topology proof, some of them require lots of other machinery than algebraic topology, but all of them can be understood with deep clarity by applying algebraic topology.

• The trefoil knot cannot be unknotted without cutting it.
• A polynomial of degree $d$ with complex coefficients has exactly $d$ roots, counted with multiplicity (The fundamental theorem of algebra).
• Given $n$, there is a field structure on $\mathbb{R}^n$ with continuous field operations if and only if $n=1,2$. Furthermore, there is a division ring structure (i.e. like a field but allowing multiplication to be noncommutative) if and only if $n=1,2,4$.
• The 2-sphere in $\mathbb{R}^3$ can be turned inside out without creating any creases, as long as you allow it to pass through itself (Smale's sphere eversion).
• There exists a compact 7-dimensional smooth manifold which is homeomorphic to $S^7$ but is not smoothly homeomorphic to $S^7$ (Milnor's exotic sphere).

One could extend this list a long, long time.

Answer 2

When I was a graduate student (in the late 1950s) I was advised to read the Introduction (and nothing else!) of Solomon Lefschetz's "Introduction to Topology".

I think the point is that to understand what is going on in Algebraic Topology today you have really to understand the history, and how the subject arose. But this is usually neglected in the books, which tend to present what the author regards as the current "canonical view". There is an excellent (but expensive) book "History of Topology", edited by I.M James.

Einstein argues in this article that it is important to ask fundamental questions on a scientific topic, as the questioner is doing. There is other discussion of "issues" here. The 2014 presentations at Paris and Galway on my preprint page discuss "Anomalies in Algebraic Topology". A discussion of anomalies should help to further the aims suggested by Einstein:

"It is therefore not just an idle game to exercise our ability to analyse familiar concepts, and to demonstrate the conditions on which their justification and usefulness depend, and the way in which these developed, little by little..."

Note that May's "Concise" book describes my book, now Topology and Groupoids, as "idiosyncratic". My aim was to find the "right" concepts to express many basic intuitions and results in algebraic topology, and this turned out in my view to require a systematic use of groupoids.

Answer 3

Topology immediately poses a question:

How can I characterize spaces, up to homeomorphism? More specifically, how can I tell if two spaces are homeomorphic?

This happens because homeomorphisms are the structure-preserving maps of topological spaces, so we are interested in being able to determine when two spaces are "the same thing".

Now, we have compactness: I know that $S^1$ is not the same as $\mathbb{R}$, since "compactness" is invariant under continuous maps. It is arguably the first topological invariant we encounter.

We also have connectedness: I know that one ball is not the same as two separated balls, since one space is connected and the other isn't. I also know that $GL(n,\mathbb{R})$ is not the same as $\mathbb{R}^{n^2}$. Diving a bit more, we also know that $[0,1)$ is not the same as $\mathbb{R}$, since taking $0$ out of the first still leaves a connected set, but taking one point out of $\mathbb{R}$ separates it into two connected components.

Okay, we can also say that $\mathbb{R}$ is not the same as $\mathbb{R}^2$ for a similar reason: taking a point out of $\mathbb{R}$ separates it into two connected components, and taking a point out of $\mathbb{R}^2$ is still connected.

But, unfortunately, those tools are not so powerful as we want. How would you argue that $\mathbb{R}^5$ is not the same as $\mathbb{R}^6$?

Algebraic topology gives us better tools to answer these questions (although it is a homotopy-type "tester" a priori).

Not only that, but homotopy and homology are clearly related to a notion of "holes" in a given space: a notion that is very geometric.

An interesting highly non-trivial application of algebraic topology is Morse Theory. Essentially, it allows you to estimate the number of critical points of a given well-behaved differentiable function of a manifold using the homology of the manifold, and vice-versa.

Answer 4

Higher dimensional spaces! That's my answer.

You can use intuition for lower-dimensional spaces, like the topological space of complex numbers, although even there, intuition often gives the wrong answer. But if you try to work out even plausible conjectures as to global topology in higher-dimensional spaces, intuition fails completely.

If the word "application" is important, as opposed to just the mathematician's desire to understand things, then one answer to that (amongst very many) lies in Lie groups. Quantum field theories are very strongly related to properties of Lie groups, but such groups have quite complicated higher-dimensional topology. Just knowing the Lie algebra of a Lie group doesn't tell you the global structure. To go from local to global topology, you need tools and machinery, and algebraic topology provides a very powerful tool-kit for investigating global topology.

In short, algebraic topology gives certainty and clarity of understanding of global topology in higher-dimensional manifolds where the intuition cannot go. That's just my opinion anyway.

Of course, then you would have to ask why anyone would want to know what the global topology of a topological space looks like. But that's actually a different question to what you are asking. Algebraic topology just gives you the tools to investigate the global topology.