Algebraic Topology 22: Cup Product of Torus & Klein Bottle

We calculate the homology, cohomology, and cup product of the torus and Klein bottle. For the Klein bottle, we calculate the cohomology and cup product using Z_2 coefficients.

Presented by Anthony Bosman, PhD.

[Helmutandmoshe]

Is this lecture series continuing soon? I hope so! It has been really great.

[elainezhang]

Thank you very much for your lecture series! You absolutely gave me the intuition and hands-on experience quickly in algebraic topology which I have been eagering for a long time. Please please keep doing it!!!

[algebraist_]

I would like to know if the course continues in the autumn semester? I would be very happy with it. It would be nice to go through the Hatcher book.😊

[-minushyphentwo]

I paused every time you were about to calculate something (the cohomology and cup product on the torus, the same for the Klein bottle in Z coefficients, then with Z2 coefficients), which was a good exercise. This time I actually understood what was going on! (unlike the first time with homology)

[-minushyphentwo]

When I clicked on this video I thought you were going to multiply the torus and the klein bottle using the cup product, but then I realized it actually meant “evaluating the cup product on the generator cycles of the cohomology groups of the solids”

[timelsen]

Thanks to you for explaining such confusing material so well!

[xanderlewis]

Just a quick off-the-cuff comment after seeing cohomology for the first time: since elements of the chain groups are Z-linear combinations of the generators (same thing as maps from the generators to Z), and elements of the cochain groups are homomorphisms from the free groups on the generators into Z (same thing as arbitrary maps from the generators to Z) and the addition operations on each coincide, aren’t they each (at least for Z coefficients) the same? I guess they might be, but the homology can differ because the boundary maps might differ. Also, I guess this has something to do with why the case of cohomology with Z coefficients is a special case and why in the torsion-free case they actually are the same.

[ompatel]

Amazing video professor also expecting a course in intersection theory

[Desidarius_Erasmus]

Sir I am watching you from India . You have done an extraordinary attempt to help us to overcome fear of algebraic topology .
Sir this is my humble request to you please discuss about nets and filters too . This is also pretty confusing .

[SnackFatson]

Does this have to do with colliding particles in gravity waves data?

[fanalysis]

Maybe there could be a follow up on intersections

[md.mehedihasanrasel]

Need next lectures sir. Eagerly waiting for that sir.