What is...the cohomology ring intuitively?

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

This time.
What is...the cohomology ring intuitively? Or: Counting intersections.

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.
http://www.dtubbenhauer.com/youtube.html

Website with exercises.
http://www.dtubbenhauer.com/lecture-a...

Cup products.
https://en.wikipedia.org/wiki/Cup_pro...
https://ncatlab.org/nlab/show/cup+pro...
http://www.math.toronto.edu/~drorbn/c...

Cup product and cellular cohomology.
https://mathoverflow.net/questions/42...

Cup products and intersections.
https://math.berkeley.edu/~hutching/t...
https://mathoverflow.net/questions/35...
https://mathoverflow.net/questions/20...

Cup product of the torus.
https://math.stackexchange.com/questi...
https://math.stackexchange.com/questi...
https://math.stackexchange.com/questi...

Cohomology and its ring structure.
https://en.wikipedia.org/wiki/Cohomol...
https://en.wikipedia.org/wiki/Cohomology

Transversality.
https://en.wikipedia.org/wiki/Transve...)

Pictures used.
https://en.wikipedia.org/wiki/Torus#/...
https://en.wikipedia.org/wiki/Fundame...
https://en.wikipedia.org/wiki/Transve...

Hatcher’s book (I sometimes steal some pictures from there).
https://pi.math.cornell.edu/~hatcher/...

Always useful.
https://en.wikipedia.org/wiki/Counter...

Mathematica.
https://mathworld.wolfram.com/CupProd...
https://mathworld.wolfram.com/Cohomol...

#algebraictopology
#topology
#mathematics