Covering spaces | Algebraic Topology 26 | NJ Wildberger
Insights into Mathematics
41,894 views
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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12 Comments
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
Covering spaces | Algebraic Topology 26 | NJ Wildberger
Insights into Mathematics
41,894 views
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.
************
Research Gate page: https://www.researchgate.net/profile/...
Blog: http://njwildberger.com/
Online courses at openlearning.com (currently Algebraic Calculus One): https://www.openlearning.com/courses/... Please join us for an exciting new approach to one of mathematics' most important subjects!
Patreon: Patreon: njwildberger Your support would be much appreciated.
Wild Egg Maths YT channel: https://www.youtube.com/channel/UCriF...
Insights into Mathematics Playlists:
The Algebra of Boole, Logic and Circuit An... (31 videos)
Box Arithmetic: a new framework for Mathem... (18 videos)
Hypergroups and Diffusion Symmetry: an int... (6 videos)
Rational Trigonometry for maths, physics a... (4 videos)
Sociology and Pure Maths (44 videos)
Old Babylonian mathematics and Plimpton 322 (8 videos)
Math Foundations (226 videos)
Math Seminars N J Wildberger (26 videos)
Math History (ancient to modern) (45 videos)
Geometric Linear Algebra (43 videos)
Algebraic Topology (40 videos)
Universal Hyperbolic Geometry (55 videos)
Differential Geometry (34 videos)
Elementary Probability and Statistics (8 videos)
Math Terminology for Incoming Uni Students (9 videos)
Famous Math Problems ( 46 videos)
Elementary Mathematics Explained (K-6) (40 videos)
Ancient Mathematics and insights of Howard... (7 videos)
Wild West Banking: A mathematician goes We... (7 videos)
Playing Go: the ancient oriental board game (19 videos)
Maths and Music (21 videos)
Year 9 Mathematics (review fractions, deci... (10 videos)
Wild Trig: An introduction to Rational Tri... (94 videos)
Wild Egg Maths Playlists:
Intro to Algebraic Calculus with Box Arith... (4 videos)
Classical to Quantum (for Members) (64 videos)
Solving Polynomial Equations and the Geode... (45 videos)
De Casteljau Bezier curves and associated ... (20 videos)
Exploring q-series (for Members) (8 videos)
Six: A mathematical exploration (9 videos)
Algebraic Calculus One: a new foundation f... (52 videos)
Advice to prospective research mathematici... (9 videos)