An introduction to homology | Algebraic Topology | NJ Wildberger

We briefly describe the higher homotopy groups which extend the fundamental group to higher dimensions, trying to capture what it means for a space to have higher dimensional holes. Homology is a commutative theory which also deals with this issue, assigning to a space X a series of homology groups H_n(X), for n=0,1,2,3,....

In this introduction to the subject we look at a particular graph, discuss cycles and how to compute them, and introduce the first homology group, admittedly in a rather special restrictive way.

We then generalize the discussion to a general graph, using the notion of a spanning tree to characterize independent cycles in terms of edges not in such a spanning tree.

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[oscarwal]

that's how math should always be taught, the down to top approach with lots of examples and easy to remember catchphrases is great. thank you !

[bogdandobondi]

I might not share your aversion towards the reals (or infinite sets), sir, but besides that this series of lectures contains perhaps the most useful videos on Youtube. I have great professors but they all have teachingstyles which differ from yours. And in this difference lie the much appreciated benefit for me. Thank you very, very much for the entire series and not only this lecture.

[aaos]

The best day of my life. too often I saw it without a proper introduction.

huge and grateful thanks for the whole course

[shilangyu]

This is the best introduction to homology I have ever seen. I had a hard time grasping this topic, books often skips some small (yet important) details. Watching this cleared everything up. Thank you, your style of teaching is very approachable.

[dhaka_mathematical_school]

Professor N J Wildberger, you are one of finest teachers on this planet.

[polopadic]

perhaps a perfect lecture, or as close to perfect as it gets

[michelhenridevoret]

Amazing series of crystal-clear lectures on a difficult and complex topic. Prof. Wildberger is a mathematical pedagogy genius. This is the best introductory material on 20th century mathematics that I have ever seen on the web so far, and particularly on YouTube. I will follow any new series of lectures that Prof. Wildberger will put on YouTube!

[nidhirathi]

Dear Sir, you have a gift of teaching in a wonderful way. Highly appreciate it. Keep making videos of whatever classes you teach. Thank you so much. I have seen the whole series.

[Crasshopperrr]

I definitely appreciate whoever edited out some segments! Shorter is better!! =)

[TenzinLundrup]

I heard the term Khovanov homology in a lecture by the string theorist Ed Witten (also winner of the Fields Medal) so here I am as a layman learning some basics which Witten must also have had to learn at some point. Thank-you Prof. Wildberger. p.s. Intuition comes from examples (I would imagine) not abstract definitions and relations. Also, the structure of space in LQG (Loop Quantum Gravity) is all about vertices and edges; see the lectures by Rovelli on youtube. Spacetime is becoming relational and discrete in this description.

[clearclearskies]

Your love for the subject really shows in THIS lecture. Beautiful stuff. Thank you very much.

[alvaroballon]

Thank you! I was trying to learn this by myself but I was failing miserably. Now Im getting it so easily

[dukeyin]

The best down to earth, intuitive, soft intro to the subject

[dirkdugan]

I'm so glad this series was posted. Unfortunately, my professor has the tendency to verbalize most of the details, and also to avoid writing coherent sentences. So this is really helpful.

[AnnevanRossum]

What a great lecture! The difference between homotopy and homology is made intuitive already at the start. The lecture itself is very easy to follow. Even doing the matrix elimination step by step. It's so nicely self-contained that I didn't mind that it could be summarized. I'll definitely check out more of his lectures!

[erelsgl]

What a wonderful talk. I tried to study from Wikipedia and it was so complex, but your talk make it seem so simple! Thanks a lot.

[bohu]

It was very helpful starting from a simple example and go to more general definitions step by step. Thank you for making this amazing series!

[xy]

A useful lecture. I would be also helpful to have some exercises, additional material supplied with it. I searched your university website, but could not find anything of that kind.

[njwildberger]

Yes, all higher homotopy groups turn out to be commutative, but this is not entirely obvious, is not part of the definition (ie they are defined as non-commutative groups), and doesn't apply to the first homotopy group (the fundamental group). The homology groups however are already defined in the framework of commutative groups. So I think the statement is still valid.

[cristafractal]

Thank you so much for posting this lecture, it is a nice and intuitive way of understanding homology groups.