Algebraic Topology 1: Homotopy Equivalence

When are two shapes the "same"? Topics covered include deformation retract, homotopy of maps, and the homotopy equivalence of spaces.

Presented by Anthony Bosman, PhD.

[MathatAndrews]

CORRECTION: The very last lemma (time 1:04:00) stated that if you have a quotient map X —> X/A, then X and X/A are homotopy equivalent. I forgot to add that this is only true if the subcomplex A is contractible (homotopy equivalent to the point).

Without the requirement that A is contractible, the lemma may not hold. For example, the circle S^1 is a subcomplex of the disk D^2, namely its boundary. Therefore we can consider the quotient map D^2 —> D^2/S^1. The image of this map is itself just the sphere S^2 (think about gluing all the points along the boundary of a disk together to give you a single point — the point becomes a pole of the sphere). But D^2 is *NOT* homotopy equivalent to S^2. (Why not? One is contractible and the other is not.)

Why doesn't the lemma work here? Because we quotiented out by the subcomplex S^1 which is not contractible violating the requirement.

[imthebestmathematician]

Many many thanks to you professor Andrew, you are truly a gifted professor

[ormerodp]

This is awesome, thanks so much! Looking forward to future videos. Keep up the great work.

[abdelazizmegdiche]

Hi, just discovered this channel and loving it so far! I'm self-studying algebraic topology with Hatcher's book, and solutions for the exercises are notoriously hard to find. Could you go through some of the exercises for each chapter ? On a personal note, I'm stuck on some of the first few exercises of chapter 1.1 - namely exercises 1, 4 and 6. Also, some category theory might be beneficial to students. Once you get the hang of it, I find that it helps put things together in a way that reduces a lot of the mental load for AT.

[Zeitaluq]

Brilliant lecture. Really useful with a large clear pad to sketch these concepts. The 'north' and 'south' pole of the sphere finally touching as the string shrinks and disappears. Seemed to 'click' conceptually, very good introduction.

[jayloncarter]

This lecture is a lot better than the one by pierre albin

[isaiasferrer]

Thank you very much for this! I have a question. I am interested in the application to learn physics, do you think the course and the book of Hatcher would be a good approach? Thanks!!

[kapoioBCS]

Why there are so many ads inside this video, it is like every 3-5 minutes 😢

[vekyll]

How exactly is R2 contractible? Or D2\S1? The problem for me is not that it's unbounded, but that it doesn't have a boundary. If f0 is mapping all R2 to (0, 0), while f1 is the identity on R2, don't {fi((0, 0)):i@[0, 1]} have to be closed?

[joshuad.furumele]

46:18
Conjecture: if X and Y are homotopy equivalent, then there is a continuous image of Y (resp. X) that is a deformation retraction of X (resp. Y).

This is prolly true... is the converse true?

[dr.aaqibiqbal]

i think you made a mistake in the definition of homotopy ( at 14:00)

[hyperduality]

Retraction (convergence, syntropy) is dual to inclusion (divergence, entropy).
"Always two there are" -- Yoda.
Attraction is dual to repulsion -- forces are dual!