Algebraic topology: Fundamental group

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This lecture is part of an online course on algebraic topology.

We define the fundamental group, calculate it for some easy examples (vector spaces and spheres), and give a couple of applications (R^2 is not homeomorphic to R^3, the Brouwer fixedpoint theorem).

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DavidMcKnight

I'm addicted to this channel! it's so good! thank you Prof. Borcherds!

adelvoid

Insight/Time is maximized in these videos, best lectures I've found on YouTube

onlyapawnintheirgame

Really nice how accessible these lectures are!

TribeWars

In the proof of Brouwer's theorem, you can instead get the direction from f(x) to x and then extend the line to where it intersects the boundary circle. This way you do get the identity map when you restrict to S^1.
Also, I find it funny how the contradiction in this theorem boils down to S^1 not being a retract of B^2, and the contradiction there just boils down to 0 != 1.

f-th

This is great. I'm only recently after reading about this in Mendelson + Gamelin's books. Thank you professor for continuing on with this series & this channel.

shingles

At 18:19, you say that contractibility of the unit interval is the reason we can find those finitely many closed intervals with image contained in one disc. Aren't we using compactness rather than contractibility though? (the preimages of the discs are open, so we can find finitely many open intervals that cover the unit interval and from those we can find the a_i)

Also why do we need 4 "discs" to cover S^2? Wouldn't two suffice? (S^2 -- northpole and S^2 -- southpole)

Joker-rvtd

At 14:20, how do we prove the forward direction of the claim, that is, if the two maps are homotopic then their lift to R must have the same image of 1?

ritabrataray

Do you need algebraic topology for physics

davidaugustyn

Did you learn algebraic or differental topology first

davidaugustyn

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