algebraic-topologyapplicationsbig-list
What are some nice applications of algebraic topology that can be presented to beginning students? To give examples of what I have in mind: Brouwer's fixed point theorem, Borsuk-Ulam theorem, Hairy Ball Theorem, any subgroup of a free group is free.
The deeper the methods used, the better. All the above can be proved with just the fundamental group. More involved applications would be nice.
When Poincaré first envisioned algebraic topology, he envisioned it as a study of smooth manifolds under the equivalence relation if diffeomorphism [Analysis situs, pages 196-198]. Lefshetz [Topology, Amer. Math. Soc. Colloq. Publ 12 (1930), page 361] wrote that Poincaré had tried to develop the subject along `analytic' lines, but had turned instead to combinatorial methods because the analytic approach failed for example in the Poincaré duality theorem.
Algebraic topology developed in the PL category (Combinatorial Topology), because it was believed that this would give a useful avenue of attack on the differentiable case. Great algebraic topologists of the early 20th century (Reidemeister, Seifert, Schubert$\thinspace\ldots$) all worked with triangulated PL manifolds, and wrote good, precise, rigourous papers which are still valuable today (I think that Schubert's Topologie is one of the greatest topology textbooks ever written). They were gluing together, and subdividing, finite collections of simplices; their group theory was combinatorial; and the subject as a whole had a highly combinatorial flavour to it. And it was great! Everything was explicit, and there was no need for fudgy handwavy `corners can be rounded' type arguments to be thrown around. In my opinion, simplicial complexes continue to be the best setting to work explicitly with linking forms, for example.
In the 1950's and 1960's, with work of Smale, Thom, Milnor, Hirsch, and others, honest smooth algebraic topology became possible, and the relationship between PL and smooth categories was clarified. And after that, people began switching back and forth at will when it was possible to do so, and, with the basic groundwork for algebraic topology established in both categories, the combinatorial flavour of the subject became dulled. Combinatorial Group Theory went off and became its own subject, and the majority of topologists no longer saw the need to mess about with explicit triangulations of manifolds- they just worked directly with invariants of the chain complex. And CW complexes became used instead of simplicial complexes, for example because the dual cell subdivision of a simplicial complex need no longer be a simplicial complex.
But "combinatorial topology" in its former sense still very much exists. An it's not going to go away. To programme topology into a computer for example, you need an explicit triangulation, and the work is all combinatorial and PL. See for example Matveev's Algorithmic topology and classification of 3-manifolds. The constructivist argument would be that `real world' manifolds (whatever that means) are PL.
Answers to this sort of question obey an uncertainty principle: The more precisely I outline a manageable "syllabus", the less likely it is to match your specific needs and interests. That said, here's a rough outline based on Hatcher's Algebraic Topology that can be adjusted as needed.
General advice: Most of Hatcher's sections begin with a paragraph or two regarding the usefulness of or intuition for the topic at hand. I strongly recommend revisiting these gems of wisdom as you move through the text; they offer a review of the central ideas and a sort of road map for the subject. Also, given the volume of new ideas, I emphasize the standard advice of skimming proofs during an initial read of each subsection and then going back to understand the details of the arguments. Finally (but first, really), read the preface! It's an instruction manual for how to read the book and understand its place in the broader literature.
Chapter 0: Some Underlying Geometric Notions. Read the whole thing at first, but don't expect yourself to memorize the details of every construction in one read. Since it sounds like you know the basics of homotopy of maps, homotopy equivalence, contractibility, et cetera, your goal here should be to make sure you're comfortable with CW complexes and their homotopy properties (e.g. homotopy equivalence under collapsing contractible subcomplexes, equivalence of homotopic attaching maps, CW pairs possessing the homotopy extension property).
Exercises: 1-6, 10, 17, 19
Chapter 1: The Fundamental Group. If you feel pretty good about fundamental groups, van Kampen's theorem, classification of covering spaces, just skim $\S$1.1-1.3 and review the statements of theorems/propositions/corollaries and definitions typeset in bold. Read the examples and "Applications to Cell Complexes" in $\S$1.2, though you can skim the more detailed discussion/examples of covering space actions on a first read. The appendices $\S$1.A-1.B are fun and have plenty of applications but are not strictly necessary. (But do read the definition of a $K(G,1)$ and statement of Proposition 1B.9.)
Exercises: $\S$1.1 -- 5, 6, 8, 10, 13, 16; $\S$1.2 -- 4, 6, 8, 9, 10, 15, (+22 if you like knots); $\S$1.3 -- All great, so do what time permits.
Chapter 2: Homology. Almost nothing to skip in $\S$2.1. Many people skim the details of barycentric subdivision/Proposition 2.21 in the section on the Excision Theorem, but you definitely need to understand the statement and the way it is used to prove the main theorem. Lots of courses skip many of the examples in $\S$2.2. You should read about cellular homology, but your goal should be computational intuition and not mastery of the details, such as why it is equivalent to the other theories. Learn Mayer-Vietoris very well, homology with coefficients well, and get a feel for Euler characteristic. Some people glaze over $\S$2.3, but it helps you organize your thoughts about homology and provides an introduction to category theory -- something you need sooner or later. For $\S$2.A, at least internalize the central statement: $H_1(X)$ is the abelianization of $\pi_1(X,x_0)$. Many courses skip $\S$2.B-2.C.
Exercises: $\S$2.1 -- 4, 5, 7, 11, 13, 15, 16, 20, 22, 27, 29, 30; $\S$2.2 -- 4, 12, 14, 15, 31, 32, 41; ($\S$2.3 -- Try all, if you read the section.)
Chapter 3: Cohomology. In a first reading of $\S$3.1, you can skip the details of the discussion on pages 191-195 beginning after "Our goal is to show that the cohomology groups..." and ending before "Summarizing, we have established...". For $\S$3.2, some people only learn the statement of the Künneth formula during a first reading. Skip all but the first theorem in "Spaces with Polynomial Cohomology" if you'd like. As for $\S$3.3, there's much debate about how much detail a first reading about Poincaré duality should include. I suggest reading all of the introduction and "Orientations and Homology", plus all of "The Duality Theorem" up to (and including) the statement of Poincaré duality for closed manifolds. Also learn the results (but not necessarily the proofs) in "Other Forms of Duality". For $\S$3.A, just learn the major results and computational propositions/corollaries. Same, but less so, for $\S$3.B. Skip $\S$3.C-3.E and $\S$3.G-3.H for now, but maybe check out $\S$3.F for limits.
Exercises: $\S$3.1 -- 5, 6, 8, 13; $\S$3.2 -- 1, 3, 7; $\S$3.3 -- 2, 3, 5, 7-10, 16, 24, 30, 31, 32, 33
If you have time: Learn the basics of homotopy theory. Try to read $\S$4.1 (which is full of really useful ideas), plus some of the other fundamentals like the Hurewicz theorem, fibrations and fiber bundles, and the connection between singular cohomology and Eilenberg–MacLane spaces.
Best Answer
This is my favorite. One can show that for any continuous map from $S^{1}$ to $R^{3}$ there is a direction along which the map has at least 4 extrema (in particular, at least 2 global minima and 2 global maxima.) More colloquially, one can show that every potato chip can be placed on a table so its edge touches the table in at least two points and its edge simultaneously has two points of maximum height.