Algebraic Topology 10: Simplicial Homology

We discuss higher dimensional homotopy groups, highlighting the difficulty of calculating them for even spheres, motivating the shift to homology which is easier to calculate. Then we define simplicial complexes, and the more general delta-complexes, and show how we use these to create a chain complex with a boundary operating relating the abelian groups with basis the n-simplicies. We give several examples of calculating simplicial homology, including the disk, circle, sphere, and torus.

Presented by Anthony Bosman, PhD.

[jeffreyjones]

An One hour and a half video is better than my Top2/Alg Top prof ever could be.

[akashsudhanshu]

thankyou so much, your lectures are very clear, easy to grasp. it really helped me.

[ishaanivaturi]

Great lecture! I think it may have been cool to see two different representations of S^1 for example, and show that they still produced the same results. I tried it on my own for practice and to verify, which I think was a good exercise

[algebraist_]

Kedves Professzor úr! Én messziről, Európából, Magyarországról követtem az Ön briliáns előadásait. A Hatcher könyvet is olvasom párhuzamosan, természetesen. Nagyon várom a következő előadásokat. Az elmúlt több mint két hét jó volt az ismétlésre, de már ideje a továbbhaladásnak. Remélem hamarosan itt lesz az újabb lecke.😊😊😊

[-minushyphentwo]

1:06:37 Homology groups of the sphere and the torus

[diegoangulo]

That is the most epic sweater viz ever

[temp]

Excellent lecturer and material. I find the Hatcher book detailed but uninspiring

[コンティオプル]

H2(S2) left me slightly confused with regards to orientation. I understand that because the outward surface consisting of T and B is connected, the orientation should stay the same, so we use the right hand rule to make sure of that. However, what if I were to construct a sphere that has one more D2 glued to the rims, splitting the interior into two "cavities"? (A shape that seems homotopy equivalent to a S2 v S2) With regards to connectedness and holes, the cavities look exactly symmetrical, but if the surfaces are oriented, they aren't symmetrical. How am I supposed to represent that with simplices when calculating homology groups?

[erenuyank]

In minute 35, is the S1 represented as simplicial complex? Two edges intersect at two points, not single simplex

[davidhand]

I don't understand how all these delta basis form groups. What are the points in delta 0 acting on, and what is the operation? What's the inverse and identity? What does it mean to multiply them by integers, and is that different from exponentiation in the normal group sense?

Why are they all Z and not R? I'm so lost on this one.

[-minushyphentwo]

I followed along with the method of computing homology groups, and then counted the cycles in the surface, which matched. Now I know how to find the groups, and part of what they mean, but no idea how it works or why the quotient of the kernel by the image must be taken in particular. I feel like how I did when I learnt how to differentiate a function

[akashsudhanshu]

Like if you also want more lectures. Dear Prof. please upload further lectures after 11.

[metarestephanois]

Why are these groups well dedined ? I mean there are many ways to glue together lines to ontain a circle.

[sayanthokdar]

Sir, could you please upload a lecture on "Singular homology groups" before this coming Thursday ? it is my humble request to you, please sir . Most needed.

[ompatel]

Can you please elaborate a little how about modding out these abelian groups

[ompatel]

Please try to answer that question as soon as possible because my exam is coming up