Classification of 2-manifolds and Euler characteristic | Differential Geometry 26 | NJ Wildberger

We describe the important classification of compact, oriented 2-manifolds, and the relation with the topological invariant called the Euler characteristic. The idea is to work combinatorially, by decomposing a 2-manifold into polygon pieces which are glued, or identified, along common edges, and then performing cut and paste operations to try to get such a configuration into a normal form.

This was worked out by Dehn and Heegaard around 1910, and the resulting classification is one of the most important foundational results in differentential geometry and algebraic topology. In fact they worked in a more general context which also includes non-orientable surfaces, such as the projective plane or Klein bottle, but for differential geometry it suffices to work with orientable surfaces, which have a consistent notion of positive orientation on small circles around the surface.

Remarkably, the resulting classification shows that the Euler invariant completely determines the compact oriented surface.

NOTE: Due to a mistake in numbering, there is no DiffGeom27 video! So please skip ahead to DiffGeom28.

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[saradanhoff]

Given the depth at which you teach your subject, to such a broad range of experience levels of students, and still provide breathtaking insights in such an absurdly simple way, is nothing short of a miracle. You are one of the best educators in the world, Professor Wildberger and I wish I could have had the priviledge of being one of your students. Just remarkable. Thank you for providing all of us who will never meet you in person a chance to experience this. Your videos have helped change how almost all of us who have been through the whole journey you provided see the world, on a fundamental level.

[alexandra-stefaniamoloiu]

With you I redescovered the pleasure of studying math.
Your lectures are simply great!
Thank you!

[ddorman]

Fantastic!! Thank you to all the students attending these Lectures (as to the 2 students who turned in late papers, really?) I look forward to our future collaborations, peace and love, Doug.

[HDQuote]

This is what I found most interesting in this video:
V-E+F is unchanged if:
- we subdivide an edge into two edges by adding a new vertex in the middle
- we add a new Edge across a face, because we will have two new faces, but removed the old face (one new face in total) and we added a new edge.

Now when given two coverings made out of polygons, one can always find a common subdivision by adding vertecies and edges and faces, which leaves V-E+F invarient in each step. Therefor two coverings have the same V-E+F

[CJSH]

you are master!
high level than all China teacher and professor!

[FranciscoCaramello]

@21:20 how could the sphere appear if you assumed all the vertices are the same?

[michaelahlbeck]

At the end of this video you referenced a video "DiffGeom27: Problem discussion 1" but I can't find it. Am I missing it? Also, your many videos are very helpful and needed, please continue to record lectures.

[adykaadyka]

Dear Dr Wildeberger, thank you for the video! I have a question regarding the vertices in one of my exercises- I've done the labelling, and there only seems to be one vertex- does this mean my surface represented by the octagon is the projective plane? Cheers!

[georgeorfanidis]

Thank you very much for the lectures! This is a truly great help for me.
I can't find DiffGeom27, I can't wait to continue :)
Thank you again!