An algebraic ZIP proof of the classification | Algebraic Topology | NJ Wildberger

In this lecture we sketch an algebraic version of Conway's ZIP proof of the classification of two-dimensional surfaces. One key idea is to replace the basic polygons in the standard proof with spheres with holes in them. We introduce an algebraic notation that allows us to manipulate (ZIP) edges together between holes on spheres.

This is the 19th lecture of this beginner's course in Algebraic Topology, given by N J Wildberger of the School of Mathematics and Statistics at UNSW.

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[kabayakawa]

It is refreshing the intuitive develop of this ltopic

[clapacademyarmandnoel]

It's really nice how Conway took the confinement of the old planar technique and elevated his method into a dimension higher by using spheres, where the added room provides more flexibility.

[dehnsurgeon]

Just to clarify, is (aa)(bb)(cc)(dd) the same surface as (aba^(-1) c)(d)+(bcd^(-1) ) up to a relabelling of the edges. I.e. does (aba^(-1) c)(d)+(bcd^(-1) ) = (aa)(bb)(cc)(dd) by simplifying.