1. Classification Theorem of Closed Surfaces | NYU Shanghai Undergraduate Research Symposium

Audio - Sicheng Xing, Weilun Wang.mp3

Project Description

We have long heard the story that a donut is similar to a coffee mug. We have been fascinated by the beauty of the geometry intuition behind geometry. The classification of surfaces is a tough and classical problem in the field of topology and geometry, by researching in this problem, we gave a glance of modern mathematic and tried our best to give a very basic categorization of surfaces on an easy case.

Comments

A Mobius strip is not a closed surface. Yet you say that you can obtain a closed (non-orientable) surface by pasting a Mobius strip to a sphere. But doesn't this surface inherit a boundary from the Mobius strip component? But maybe I just don't understand how the pasting is done.

Thank you for putting up this question! This is a good question. By pasting a Mobius strip to a sphere I mean: if we take a sphere and cut one "hole" on it, then the thing that we obtain has one "boundary" (see picture below), and this thing has one edge, which is the red line in the picture. Then since Mobius strip has only one edge, then we can "paste" the edge of the cut sphere and the edge of the Mobius strip together to eliminate them both. Is this clear to you now?

Cutting the sphere so that we can paste a mobius strip on it.