User r. andrew hicks - MathOverflow

Answer by R. Andrew Hicks for What should be taught in a 1st course on smooth manifolds?

2011-03-19T01:13:58Z

I don't believe either of those books covers distributions and the theorem of Frobenius. Connections to partial differential equations in general I think are good topics.

Guillemin and Pollack is a book I like a lot, but chapters 2 & 3 (transversality and intersection) always seemed a bit specialized for a first course. Although, the title is, after all, "Differential Topology". My experience is that people tend to cover just chapters 1 & 4.

The definition of a manifold in G&P is as a subset of $\mathbb{R}^n$ (as in Milnor). As I recall the the definition of diffeomorphism is such that a cube and a sphere are considered not to be diffeomorphic. This is because G&P define a map at point of a manifold to be smooth if it can be extended to a map on an open set of the ambient space that is smooth in the sense that it is a map from an open set in $\mathbb{R}^n$ to $\mathbb{R}^m$. I never understood, or saw, how this approach can be used to think about different differentiable structures on manifolds. Since there is only one differential structure on $S^2$, the definition I mention above of diffeomorphism seems to at odds with the general one, given for example in Spivak volume 1. (If anyone could explain this to me I'd be grateful. As a student I found this confusing and still do.)

What I am getting at in the above paragraph is that an additional topic might be the general definition of differentiable manifold. It's nice have projective spaces and Grassmanians at least in ones collection of examples.

Comment by R. Andrew Hicks

2011-03-20T13:59:18Z

Jack - Spivak's "cube-sphere", as you so elegantly put it, does come equipped with a metric. So it has geometry, which makes it isometric as a metric space to a cube.

Comment by R. Andrew Hicks

2011-03-20T03:24:10Z

@Jack: My last statement "the two categories of differentiable manifolds obtained from G&P and Spivak are very different." is not correct.

Comment by R. Andrew Hicks

2011-03-20T03:02:26Z

@Jack: Given the G&P definitions the cube is simply not a differentiable manifold, as you point out, so of course it isn't diffeomorphic to the usual round $S^2$. They are however, homeomorphic. From the viewpoint of the Spivak Vol. 1 definitions you would say "they are homeomorphic and AFTER you put a (any in fact) differentiable structure on the cube then it is a theorem that it is diffeomorphic to $S^2$." If this is right then the usage is really different. (Another way to say this(?): the two categories of differentiable manifolds obtained from G&P and Spivak are very different.)

Comment by R. Andrew Hicks

2011-03-19T16:16:20Z

Thanks for that clarification. So it really is a different definition of diffeomorphism, and so perhaps is deserving of a different name, like 'ambient diffeomorphism'.