Knot Theory 5: Fundamental Group 

For those who, like me, were a little confused at the associativity part, there is something that wasn't highlighted enough (although the lecturer remembered to mention it at 34:28): the group structure is not discovered outright on the set of loops because it's a needlessly big, arbitrary and nasty object (think of the parametrization issues that prevent associativity of multiplication (27:52 to 28:45)). That's why he has to take the quotient by an equivalence relation that preserves enough topological information (homotopy) and is compatible with the multiplication of loops. That's also why he only needed to demonstrate associativity up to homotopy... Aside from that, the lecture was very clear, thank you. An African autodidact

Firstly, let me say that this lecture series is fantastic! As a secondary school maths teacher in the UK I've really enjoyed learning some properly new maths for the first time in a while, and you explain everything so clearly that I feel confident explaining the basics of knot theory to the children I teach. Secondly, for anyone else struggling to fill in the details of the Borsuk-Ulam proof, I think there are a few key points that weren't initially obvious to me that might help: 1) Under the assumption, h is a continuous map from all of S2 to S1, and since h(-x) = -h(x) it maps antipodal points on S2 to antipodal points on S1. 2) Imagine applying h to a loop on the equator that winds round at least once, and say you start at x0 on the equator (and thus h(x0) on S1). Once you've got halfway around the equator to -x0, the image of h must be exactly halfway around S1 (to -h(x0)), and since antipodal inputs give antipodal outputs h(x) can't come back the same way as you wrap around the rest of the equator. That means that the loop on the equator maps to a loop on S1 *with the same winding number*. The issue I had was seeing why it was necessarily the case that we could find loops on the equator that mapped to non-trivial loops on S1, but this is why. 3) Finally, you can homotopically deform the loop on the equator to the constant loop in S2, and since h maps all of S2 continuously onto S1 this must mean that the image of your loop is homotopically deformed onto a constant point in S1, *regardless of the original winding number*. This is clearly impossible (since the fundamental group of S1 is non-trivial), hence the contradiction. I'm looking forward to working my way through the rest of the lecture series!

This guy is an amazing teacher.

This course is totally fantastic! I've been searching for videos about topology and this is exactly what I am longing for :)

Very cool, indeed. Loved the BFPT proof.

very nice ..... we need lectures on Algebraic topology ....... you are talented in teaching.

In your proof of the BFPT, I think you missed pointing out that by construction r(x)=x on S1. You need this property to show that r ∘ ft(0) = f, i.e. that r ∘ ft is indeed a homotopy of f and not some other curve.

fzero was a great game

Also, it's not enough to prove that there is a one to one mapping of pi_1(S^1) to Z. We also need to prove, that n + m map to n circles around, and then m circles around. It's obvious, but I think it should be mentioned.

Are paths embedded in 2D or 3+D space?

Some math joke for y'all: why is Andrew always saying "x knot" if he clearly labeled it with an unknot symbol