More homology computations | Algebraic Topology 34 | NJ Wildberger

In our last lecture, we introduced homology explicitly in the very simple cases of the circle and disk. In this lecture we tackle the 2-sphere. First we compute the homology using the model of a tetrahedron: four 2-dimensional faces, but no 3-dim solid. This illustrates how linear algebra naturally arises in this kind of problem.

We then provide a much simpler alternative calculation using the more flexible framework of semi-simplicial complexes, or delta-complexes, where only two triangular faces are needed, and the calculation is much simplified, however still giving the same final result (which by the way is that H_0 (S^2)=Z, H_1 (S^2)=0 and H_2 (S^2)=Z, with all higher homology groups being 0.

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