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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 13: AlgTop12 Duality for polygons and the Fundamental Theorem of Algebra

4.1 ( 11 )

###### Lecture Details

We define the dual of a polygon in the plane with respect to a fixed origin and unit circle. This duality is related to the notion of the dual of a cone.

Then we give a purely rational formulation of the Fundamental Theorem of Algebra, and a proof which keeps track of the winding number of the image of concentric circles about the origin. This is an argument every undergraduate math student ought to know!

This is the 12th lecture in this beginners course in Algebraic Topology, given by Assoc Prof N J Wildberger at UNSW.

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