Algebraic Topology 7: Covering Spaces

We introduce the classification of covering spaces by subgroups of the fundamental group. First we look at the familiar example of the real numbers serving as a cover for the circle S^1. Then we look at other covers of the circle and see some structure emerge. We see an even richer structure when we look at the cover of the wedge of two circles, S^1vS^1. This includes a fractal as the universal cover.

Presented by Anthony Bosman, PhD.

[YitzharVered]

Probably the nost accesible way to learn algebraic topology.

[AlexHGong]

These lectures are incredibly good! I am student of Prof. Bosman now. Wish more cources on advanced topology!

[enpeacemusic]

I've noticed that the universal cover for the wedge of two copies of S_1 is literally just the cayley graph for the free group <a, b>
Is this a coincidence? Because the way you drew R as a helix (with "vertices" corresponding to the base point in S_1) is technically just the cayley graph of <a>, or Z.

[richardchapman]

Tried to do an Escher staircase style version of winding number helix going from one chirality to another. The attempt at representing an R3 object in R2 only demonstrates itself at the interstes of an even sided diagram on paper. May be fun for artwork when unable to absorb algebraic topology swiftly.

[РыжковИванСергеевич]

Thank you for an amazing course! I wonder if you have any lecture notes?

[greenland]

I dont understand the example at around minute 29, the figure on the right, with two points and 4 paths: Why can we call two paths a (respectively b) when they are not the same?

[tylerbakeman]

Great lecture. Its very clear.

He/ him is less common for referring to objects than she/ her, because relationships (ie in graph theory) are feminine by default. But, that has no affect on the topic. I just thought it was interesting.

[davidhand]

Your illustration of the one with four loops and 3 vertices has totally lost me. The vertex y0 has two arrows labeled b, one ingoing and one outgoing, on the left. How can one of them be b and the other be b inverse if they are not just one arrow? The inverse of an arrow goes back along the arrow, right? That's part of the definition of a group, right? Am I overthinking this? Does everything work if I just mentally remove one of the arrows? No, I don't think so, if b squared is a loop; what am I missing here?

Wait, wait, I think I've got it now. The inverse of b _is_ b, you just made the second arrow to make it look like a circle, showing that by0 = x0 and bx0 = y0. It still seems like this would complicate contracting some loops, though. b b inverse ought to be contractable, right?

Now I'm stuck on the fact that a does something totally different at x0 than y0. Is that just a side effect of the way you've drawn it? Like, either way, a squared does get you back to y0 or x0. Does this have something to do with the fact that we're looking at this system as a double cover of the simpler space where both a and b lead to the same point?