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# Algebraic Topology

The University of New South Wales, , Prof. N J Wildberger

Updated On 02 Feb, 19

##### Overview

Contents:
Introduction to Algebraic Topology - One-dimensional objects - Homeomorphism and the group structure on a circle - Two-dimensional surfaces: the sphere - More on the sphere - Two-dimensional objects - the torus and genus - Non-orientable surfaces - the Mobius band - The Klein bottle and projective plane - Polyhedra and Euler's formula

Applications of Euler's formula and graphs - More on graphs and Euler's formula - Rational curvature, winding and turning - Duality for polygons and the Fundamental Theorem of Algebra - More applications of winding numbers - rational curvature of a polytope - Rational curvature of polytopes and the Euler number - The geometry of surfaces - The two-holed torus and 3-crosscaps surface - Knots and surfaces I

## Lecture 3: AlgTop2 Homeomorphism and the group structure on a circle

4.1 ( 11 )

###### Lecture Details

This is the first video of the second lecture in this beginners course on Algebraic Topology. We give the basic definition of homeomorphism between two topological spaces, and explain why the line and circle are not homeomorphic.

Then we introduce the group structure on a circle, or in fact a general conic, in a novel way, following Lemmermeyer and as explained by S. Shirali.

This gives a gentle intro to the definition of a group. It also uses Pascals theorem in an interesting way, so we give some background on projective geometry.

This lecture is part of a beginners course in Algebraic Topology given by N J Wildberger at UNSW.

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