Algebraic Topology 13: Homotopy Equivalence Preserves Homology

[@-minushyphen1two379]

03:10 Recap on homotopy equivalence 06:45 First, a proof that a continuous map induces a chain map in the expected way 18:40 Proof that a chain map induces a homomorphism between homology groups in the expected way 24:50 Stating a helpful theorem: if two maps between spaces are homotopic, they induce the same homomorphisms between the homology groups 26:15 The desired result is then a corollary of this theorem 33:30 getting ready to prove the helpful theorem (which takes a lot because it requires the introduction of the prism map) 34:50 Motivating the prism map algebraically, by showing that a chain homotopy between two chain maps implies the induced homomorphisms between the homology groups are equal 45:45 Definition of the prism map by geometry At 57:15, why is the alternating sign there on the (n+1)-simplices? 57:15 Checking that the defined prism map induces a chain map

[@viniciusmiradouro1606]

Gosh, now I see why the language of Functors and Natural Transformations make things easier in algebraic topology.

[@ompatel9017]

Amazing video

[@ali-om4uv]

That was bosh amazing and a frightening reminder just how much I have forgotten 😢. I really studied alot gor that exam a while back

[@davidutoob]

I think your sign error comes from an error in the ordering of the two triangles in the top line. It should be ∂(F ... [v0, w0, w1] - F ... [v0, v1, w1]), not the other way around.

[@MathatAndrews]

Thank you!

[@コンティオプル]

Got through the weeds! Thanks for the lecture!

[@コンティオプル]

26:55 The f and g introduced in the corollary and used in its proof, are different from the f and g of the theorem. In the corollary, their compositions are homotopic to identity, whereas in the theorem, they themselves are homotopic with each other. Might confuse someone.

[@MathatAndrews]

That's true. I hope it wasn't too confusing.

[@コンティオプル]

At 30:23, isn't the composition of f ∘ σ = f#, not f*?

[@MathatAndrews]

It is f_# on C_n but this induces f_* on homology.

[@コンティオプル]

@@MathatAndrews Thank you!