What is...homology intuitively?

Goal.
Explaining basic concepts of algebraic topology in an intuitive way.

This time.
What is...homology intuitively? Or: What is a hole?

Disclaimer.
Nobody is perfect, and I might have said something silly. If there is any doubt, then please check the references.

Disclaimer.
These videos are concerned with algebraic topology, and not general topology. (These two are not to be confused.) I assume that you know bits and pieces about general topology, but not too much, I hope.

Slides.

Website with exercises.

Homology.

Holes.

Torus and solid torus.

Pictures used.

Hatcher’s book (I sometimes steal some pictures from there).

Always useful.

Mathematica.

#algebraictopology
#topology
#mathematics

[Ayan-rhgj]

Sir please never stop. You don't know how much social service you are doing. These videos are gold mines

[igragarirg]

Hello, I'm a Math student doing my end of degree thesis about De Rahm's Theorem and your videos are being incredibly useful for me to understand homology and cohomology theory. Thanks for all the content and keep up this good work!

[kienhuynh]

As a computer science student who is learning some bits of math here and there, thank you so much for this!

[shashvatshukla]

Love the enthusiasm for clear example-driven explanation.

[fieryflowers]

Brilliant! Reflects all the time that you must have put in to learn this thoroughly yourself.

[King_Illuminaughti]

Very helpful and confirmation of what I have grasped intuitively & tried to convey to a wider audience; regarding the discrete properties of mathematics & physics❣️🏆🙏🏻🗽👼🏻

[feraudyh]

When trying to learn about homology, I come up with a certain number of obstacles:
There are various theories of homology which are not quite the same.
There are various definitions of simplexes (abstract or not).
The idea of a formal sum of simplexes is not obvious to interpret geometrically. In short it's hard to know if one should associate a geometrical intuition with an abstract object.

[Jaylooker]

Homology sounds a lot like homotopy that also has (linear) representations noting cycles/loops, null homotopy, contractibility (to a point), equivalence relations, R^n vector spaces. There is Hurewicz Theorem which describes a map between homotopy groups and homology groups. Maybe this describes that, too.

[jmw]

Philosopher: "Is a hole something or a lack of something else?"
VisualMath:

[조성재-ig]

Sir, your videos are awesome! You just made me a man.

[TheTessatje]

Thanks, this was of great help! One quesiton: A group is a set and operation (fulfilling certain properties). What is the set and what is the operation in homology groups?

[mkg]

Brilliant videos....thank you so much for the kind efforts...

[axog]

I don't understand what the notation Q{x, y, z} and Q{a, b, c} means. Could you try to explain it a different way to me?

[iskandermukatayev]

it's great! thanks for the best explanation