In this video I teach the amazing @TomRocksMaths a little bit of algebraic topology, specifically the fundamental group. Tom also taught me some really cool fluid dynamics and you can find our collab over at his channel here:
►►► • Helmholtz Principle and Vortex Motion...
0:00 What is Algebraic Topology?
4:01 The alphabet to a topologist
8:20 The algebra of loops about a ring
14:50 Defining Homotopy Equivalence
18:54 The Fundamental Group
23:58 Fundamental Group of R^2
25:50 Fundamental Group of a Sphere
28:32 Fundamental Group of a Circle
31:45 Fundamental Group of a Torus
34:18 Proof of Brouwer's Fixed Point Theorem
We begin by talking about what connotations the words "algebraic" and "topology" have; "algebra" has a certain concreteness to it as we can add or multiply things, have explicit formulas, etc while "topology" is all about considering spaces that are thought of the same even if we continuously deformed like they were playdoh. Under this perspective, the alphabet only has three letters to a topologist, a single point, a single circle, and a double circle (the letter B).
We then define more precisely the notion of a homotopy equivalence between two maps into a space X. There is an operation we call multiplication on such paths which captures the idea of doing one path followed by the next. It turns out that the homotopy equivalence classes of loops in a space X starting and finishing from a basepoint x_0 with this notion of multiplication form the fundamental group which we often write is Pi_1(X, x_0).
Tom then computes the fundamental group of many spaces, the plane, the 2-sphere, the 1-sphere or circle, and finally - triumphantly - the torus! Finally we finish with a nice proof of Brouwer's Fixed Point theorem that uses the power of the fundamental group to arrive at a contradiction.
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