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The Princeton Companion to Mathematics by Imre Leader, June Barrow-Green, Timothy Gowers

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IV.7 Differential Topology

C. H. Taubes

1 Smooth Manifolds

This article is about classifying certain objects called smooth manifolds, so I need to start by telling you what they are. A good example to keep in mind is the surface of a smooth ball. If you look at a small portion of it from very close up, then it looks like a portion of a flat plane, but of course it differs in a radical way from a flat plane on larger distance scales. This is a general phenomenon: a smooth manifold can be very convoluted, but must be quite regular in close-up. This “local regularity” is the condition that each point in a manifold belongs to a neighborhood that looks like a portion of standard Euclidean space in some dimension. If the dimension in question is ...

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