Algebraic Topology 20: Introduction to Cohomology

We give a brief recap of homology and then show how dualizing the chain complex by Hom(--,Z) gives a cochain complex with coboundary maps that we use to calculate cohomology. We show that for finitely generated chain groups, we can calculate the cohomology in terms of the homology groups. Then we dualize with other coefficient groups G and discuss the universal coefficient theorem for cohomology.

Presented by Anthony Bosman, PhD.

[tim-cca]

I start watching your series here. I learnt the relationship of cohomology and homology. I will watch the whole series because it helps a lot and saves me some time to read books later. Good work!

[ikechukwumichael]

These series just keeps getting better and helping me understand topological data analysis and geometric deep learning. Have a wonderful Easter Sir.

[swaruppaul]

Aahh! Finally a lecture on cohomology! Really excited to learn this from you! Thank you professor ♥️

[Desidarius_Erasmus]

I am from India and have advanced topology along with Algebraic Topology in my MSc last year . Your videos are amazing and helping me a lot to overcome the difficulties in Algebraic Topology .

[Oreo_od]

This series has been helping me ton with my PhD oral exam preparations! I just hope a couple more get released before Tuesday... :D

[swaruppaul]

Finished it! It was an amazing lecture! ♥️

[AzizBouland]

How many more videos will there be in this series? I hope we get to see more of chapter 3 (and 4), these lectures are gold..

[gabesorci]

These videos are amazing. Thank you very much !!

[sayanghosh]

Much appreciated thing. Thank you Professor! 🙂

[aviralsood]

I have an additional question, how is your boardwork so clean? Do you have a rough idea of how much you are going to fit in one board? I just wing it and it ends up being very messy.

[ompatel]

Sorry professor late this time not gonna watch this video right now currently enjoying my vaccation travelling vietnam. My favourite hobbie after math is travelling. To this date i have travelled 2 countries internationally thailand and vietnam. Being in vietnam i feel like a billionare. Also thanks for the lecture in advance.

[IshouldGetQualified]

Great Video! But what does it mean for the group hom(Z, Z) to be isomorphic to Z? Aren’t they groups of maps to groups of functions? Sorry if this is a dumb question, as I have a weak abstract mathematical background

[algebraist_]

Waiting for The Sun, waiting for next lecture😊

[xanderlewis]

30:48 Yes, you can check it if you like… but you can instead just remember that contravariant represented functors preserve coproducts 😉 (a fact I learnt only very recently!).