Computing homology groups | Algebraic Topology | NJ Wildberger

The definition of the homology groups H_n(X) of a space X, say a simplicial complex, is quite abstract: we consider the complex of abelian groups generated by vertices, edges, 2-dim faces etc, then define boundary maps between them, then take the quotient of kernels mod boundaries at each stage, or dimension.

To make this more understandable, we give in this lecture an in-depth look at some examples. Here we start with the simplest ones: the circle and the disk. For each space it is necessary to look at each dimension separately. The 0-th homology group H_0(X) measures the connectivity of the space X, for a connected space it is the infinite cyclic group Z of the integers. The first homology group H_1 measures the number of independent non-trivial loops in the space (roughly). The second homology group H_2 measures the number of independent non-trivial 2-dim holes in the space, and so on.

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[marcoguitarsolo]

MASSIVE thank you for uploading this. It's so much clearer to me now.

[njwildberger]

The del-zero map is always zero, since there is no chain group of dimension -1.

[reinerwilhelms-tricarico]

I love it. Wildberger saves us from the dreary task of grinding through the usual tomes on this subject, where the authors seem to believe that drawing a single picture would make the subject somehow less glorious :-)

[ramdattjoshi]

Absolutely impressive! I am wondering if it is possible to post the videos for like: PL Gauss-Bonnet theorem, and about chain derivation results. Thank you for your generosity and your support!!!

[brydust]

njwildberger I am in an algebraic topology course that essentially starts from your 35th video (intro to homology) and we are reading hatcher which is good but I find it too verbose, are there any (modern & friendly) books you would recommend on the subject? By the way, I really have enjoyed watching these videos the subject has really come alive and I'm able to appreciate the abstract setting with your great examples.

[flagalicious]

Highly useful lectures, thanks! As I understood you used the argument that a map is injective if it is non-zero on generators. Can this always be applied in the setting of homology? I was thinking of the projection from the integers onto Z/2Z as a counterexample...

[tact]

Just perfect, thanks so much professor

[mohammadbazzi]

Thanks for the video. I would actually like to know why you considered (x z) to be equal to - c. Clearly it is the difference in orientation, but is there an argument which would work for higher dimensions?