More homology computations | Algebraic Topology | 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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[irenicity]

Very helpful for learning homology group. I have been confused by homology for a long time until now. The concrete examples are very illustrative. Thank you.

[ejigaru]

You simply saved my life! This was an awesome lecture.

[명노훈-ox]

AlgTop lectures helped me a lot. Thank you.

[johnstroughair]

Excellent lecture - cleared up things that had been confusing me. Thanks!

[alute]

This is what remedial math of any college education should start with

[mattheuspereira]

You explain very well, the first class on homology was very useful. However, for these more complicated examples, I think it's better to use singular homology. You can easily compute the reduced singular homology groups of S^n using a bunch of strong theorems.

[ivanriveraacatitla]

Just 2 class and then may be i can make my tesis, thanks