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source: njwildberger 2011年3月9日

This is the full introductory lecture of a beginner's course in Algebraic Topology, given by N J Wildberger at UNSW. The subject is one of the most dynamic and exciting areas of 20th century mathematics, with its roots in the work of Riemann, Klein and Poincare in the latter half of the 19th century. This first lecture will outline the main topics, and will present three well-known but perhaps challenging problems for you to try.

The course is for 3rd or 4th year undergraduate math students, but anyone with some mathematical maturity and a little background or willingness to learn group theory can benefit. The subject is particularly important for modern physics. Our treatment will have many standard features, but also some novelties.

The lecturer is Assoc Prof N J Wildberger of the School of Mathematics and Statistics at UNSW, Sydney, Australia, well known for his discovery of Rational Trigonometry, explained in the series WildTrig, the development of Universal Hyperbolic Geometry, explained in the series UnivHypGeom, and for his other YouTube series WildLinAlg and MathFoundations. He also has done a fair amount of research in harmonic analysis and representation theory of Lie groups.

My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/.... I also have a blog at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at http://web.maths.unsw.edu.au/~norman/. Of course if you want to support all these bold initiatives, become a Patron of this Channel at https://www.patreon.com/njwildberger?... .

**Algebraic Topology**

A first course in Algebraic Topology, with emphasis on visualization, geometric intuition and simplified computations. Given by Assoc Prof N J Wildberger at UNSW.

The really important aspect of a course in Algebraic Topology is that it introduces us to a wide range of novel objects: the sphere, torus, projective plane, knots, Klein bottle, the circle, polytopes, curves in a way that disregards many of the unessential features, and only retains the essence of the shapes of spaces. What does this exactly mean? That is a key question...

The course has some novel features, including Conway's ZIP proof of the classification of surfaces, a rational form of turn angles and curvature, an emphasis on the importance of the rational line as the model of the continuum, and a healthy desire to keep things simple and physical. We try to use pictures and models to guide our understanding.

0: Introduction to Algebraic Topology 30:01

1: One-dimensional objects 32:19

2: Homeomorphism and the group structure on a circle 52:09

3: Two-dimensional surfaces: the sphere 42:27

4: More on the sphere 40:52

5: Two-dimensional objects--the torus and genus 49:33

6: Non-orientable surfaces---the Mobius band 42:05

7: The Klein bottle and projective plane 39:42

8: Polyhedra and Euler's formula 45:35

9: Applications of Euler's formula and graphs 42:23

10: More on graphs and Euler's formula 47:50

11: Rational curvature, winding and turning 48:19

12: Duality for polygons and the Fundamental theorem of Algebra 45:36

13: More applications of winding numbers 26:59

14: The Ham Sandwich theorem and the continuum 36:26

15: Rational curvature of a polytope 50:23

16: Rational curvature of polytopes and the Euler number 35:28

17: Classification of combinatorial surfaces (I) 50:32

18: Classification of combinatorial surfaces (II) 1:00:21

19: An algebraic ZIP proof of the classification 42:23

20: The geometry of surfaces 43:54

21: The two-holed torus and 3-crosscaps surface 38:53

22: Knots and surfaces I 52:23

23: Knots and surfaces II 38:11

24: The fundamental group 43:05

25: More on the fundamental group 34:56

26: Covering spaces 53:49

27: Covering spaces and 2-oriented graphs 30:42

28: Covering spaces and fundamental groups 46:41

29: Universal covering spaces 48:17

Review: An informal introduction to abstract algebra 48:55

Review2: Introduction to group theory 46:44

Review3: More on commutative groups---isomorphisms, homomorphisms, cosets and quotient groups 32:02

Review4: Free abelian groups and non-commutative groups 50:59

30: An introduction to homology 46:57

31: An introduction to homology (cont.) 41:24

32: Simplices and simplicial complexes 49:09

33: Computing homology groups 41:07

34: More homology computations 42:55

35: Delta complexes, Betti numbers and torsion 48:17

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hi

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