Algebraic Topology 22: Cup Product of Torus & Klein Bottle

[@Helmutandmoshe]

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

[@timelsen2236]

Thanks to you for explaining such confusing material so well!

[@-minushyphen1two379]

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)

[@MathatAndrews]

Excellent!

[@alvarotejedo7868]

When will next class be available?

[@fanalysis6734]

Maybe there could be a follow up on intersections

[@Desidarius_Erasmus99]

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 .

[@algebraist_24]

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.😊

[@md.mehedihasanrasel9684]

Need next lectures sir. Eagerly waiting for that sir.

[@infiniteseries6210]

When will it continue?

[@-minushyphen1two379]

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”

[@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.

[@xanderlewis]

I just realised I’m basically just observing that finite dimensional Z-modules are all isomorphic to their duals. Which is… I guess… a very standard fact.

[@ompatel9017]

Amazing video professor also expecting a course in intersection theory

[@md.mehedihasanrasel9684]

Next lectures please dear professor

[@ikechukwumichael1383]

Much Appreciated Sir

[@abdelfattahelachab2404]

Please, the name of this teacher