Geometry Bulletin Board

I recently had the pleasure of contributing a blurb for Ravi Vakil’s book The Rising Sea: Foundations of Algebraic Geometry, which is finally being published this fall. At the start, Ravi quotes David Mumford that algebraic geometry “seems to have acquired a reputation of being esoteric, exclusive, and very abstract, with adherents who are secretly plotting to take over all the rest of mathematics.” Ravi comments that “the revolution has now fully come to pass.”

But has it?

If algebraic geometry has reshaped the rest of mathematics, why are we still using the old definition of smooth manifolds? The definition that you find in textbooks on topology and geometry is clunky: first you specify a particular atlas of charts, and then you consider an associated “maximal atlas”. Once you learn how to work with smooth manifolds, you probably never think about a “maximal atlas” again. But why put our students through this?

Here is how we should all be defining smooth manifolds. (I would be glad to hear if some textbooks are already doing this. Update: Glen Bredon’s “Topology and Geometry” (1993) gave essentially the definition I’m recommending. Georges de Rham’s “Variétés Différentiables” (1955) had something similar. Still, many newer books do not mention this approach.) I will assume that you know what a topological space is, and what a smooth (or C^∞) function on an open subset of Rⁿ is, for a natural number n. The idea, motivated by algebraic geometry and sheaf theory, is that you should characterize the smooth structure on a manifold by saying which functions are smooth.

Here is the definition. A smooth manifold of dimension n is a topological space X together with a subset S(U) (called the “smooth functions”) of the continuous functions U → R for every open subset U of X, such that:

1. the restriction of a smooth function to a smaller open set is smooth;
2. if a function f on U ⊂ X is smooth on some collection of open sets that cover U, then f is smooth on U; and
3. for every point p in X, there is an open neighborhood U of p and a homeomorphism from U to an open subset of Rⁿ such that, for every open subset V of U, the smooth functions on V are exactly the functions that are C^∞ with respect to this chart.

That’s all you need. (It is standard to assume also that the topological space X is Hausdorff and a countable union of compact subsets.) The charts in condition 3 are the same as the charts in any other definition of a manifold. The advantage of this definition is that you don’t need to make those charts part of the structure of your manifold, and then worry about when one atlas of charts is compatible with another atlas. Rather, condition 3 just asks for the existence of suitable charts.

Please feel free to use this definition when you teach about manifolds!

One side remark: there is another approach to defining smooth manifolds that makes a lot of sense for a first introduction, say for undergraduates in math. (This is used in John Milnor’s classic short book Topology from the Differentiable Viewpoint, for example.)

Namely, it is elementary to define a “smooth submanifold” of R^N. (You definitely do not need to know what a topological space is, for this purpose.) Then you could just define a smooth manifold to be a smooth submanifold M ⊂ R^N for some N. It is easy to define a smooth mapping from M₁ ⊂ R^N₁ to M₂ ⊂ R^N₂. By the Whitney embedding theorem, we know that the resulting category is equivalent to the usual category of smooth manifolds.

It’s good to be aware of this approach. But I think everyone agrees that this is not satisfying as the definition of smooth manifolds, for both practical and aesthetic reasons. The point is that you want to be able to construct manifolds in various different ways: by cutting and pasting, taking quotients by group actions, and so on. If you use the “submanifold” approach, then you would have to think about how to embed every manifold you consider into some R^N before you could even talk about it. That’s awkward. This issue strongly motivates giving an abstract definition of smooth manifolds, either in terms of atlases or in terms of smooth functions, as I’m recommending.

The revolution continues.

Image is Kawabata Gyokushō‘s Cat Seen from Behind (ca. 1887–92). Metropolitan Museum of Art. Charles Stewart Smith Collection, Gift of Mrs. Charles Stewart Smith, Charles Stewart Smith Jr., and Howard Caswell Smith, in memory of Charles Stewart Smith, 1914.