Covering spaces | Algebraic Topology | NJ Wildberger
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- čas přidán 20. 09. 2012
- We introduce covering spaces of a space B, an idea that is naturally linked to the notion of fundamental group. The lecture starts by associating to a map between spaces, a homomorphism of fundamental groups. Then we look at the basic example of a covering space: the line covering a circle. The 2-sphere covers the projective plane, and then we study helical coverings of a circle by a circle.
These can be visualized by winding a curve around a torus, giving us the notion of a torus knot. We look at some examples, and obtain the trefoil knot from a (2,3) winding around the torus.
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Yes, a trefoil knot is certainly homeomorphic to a circle: in fact any knot is. When we discuss knot theory we study how a particular circle is embedded in three dimensional space. So it is the relationship between the circle and the ambient space that is important.
NJ wildberger, you are an amazing teacher! You make my courses in algebraic topology sooo much more fun! Not just that, but you are a great teacher in general! The world benefits a lot from your works!
sir, your lectures are one of the best available online or otherwise as well
Thanks Prabhakar. If you would like to support the channel, we do have a Patreon page--from which you can then access the online Algebraic Calculus One course which is very exciting!
Excellent explained. At the moment i'm reading Czes Kosniowski book to Algebraic Topology, i like your way of teaching, really helpful. Best regards from Poland.
Yes you are right, that cover that I said was 3:1 was actually 4:1. Thanks.
Beautiful teaching! Very engaging and clear
Video Content
00:00 Introduction
00:55 Homomorphisms of fundamental groups
08:09 Covering space
21:20 Def p X --> B in a covering map if
24:40 Example 2 on images
29:29 Example 3 on images
33:30 Algebraic interpretation
36:47 Another physical model of the helical covering of a circle uses a torus - Torus knots
47:10 Example p S¹-> S¹×S¹
t-> ( 2+,3+) mod 1
50:13 Making a diagram of a torus
and a trefoil knot
52:45 Problem; Describe the torus knot associated to S¹->S¹×S¹ t->(3t,4t) mod 1
Ah... such a wonderful appearence of the fundamental group(oid) functor. I almost thought you were gonna start talking about categories haha.
Amir, if you're interested, he's quite clearly detailed his viewpoint in other videos, you need only look his name up on google to find them. But the crux of the matter is explained in MF42a: ( MF42a: Deflating modern mathematics: the problem with `functions' ) where he shows that the modern conception of functions is flawed. You can scroll through the rest of the series and find the obviously relevant videos; fair warning, you will probably not want to watch all of them in one sitting: some are quite long.
Your question is logically incoherent, since you are assuming the existence of R.
p(-3 1/3) = 1/3 mod 1
He's a realnumberphobic. ;-)
For a very good reason: the theory of ``real numbers'' is a fraud.