What are the basics of differential topology?

[Mystic998]

I was just wondering if anyone had a decent website explaining some of the basic terminology of differential topology. Specifically, I'm having a bit of trouble understanding charts and atlases and how one defines a smooth manifold in an arbitrary setting (i.e., not necessarily embedded in R^n)

 

[mathwonk]

a n dimensional manifold is a topological space, often assumed to be hausdorff, such that every point has a neighborhood homeomorphic to an open set in R^n, or one can say just that every point has a neighborhood homeomorphic to an open ball in R^n.

so we have a topological space, and an open cover of that space, and for each set in the open cover we have a given homeomorphism to an open set in R^n. This data is called an atlas, and each individual open set and map is called a chart.

the idea is that the space is the Earth and the homeomorphisms are like correspondences between points of the Earth and points on a flat paper map.

if it is possible to choose these homeomorphisms so that when two of them overlap, the composition map between open sets of R^n is always smooth, we say the atlas is a smooth atlas, and that this data defines a structure of smooth manifold on our space.

it is obvious how to define a real valued function to be smooth, by asking that be true for the appropriate compositions. the chain rule says the definitiion does not depend on which of several possible overlapping charts we use.but I believe every paracompact manifold embeds as a submanifold of R^N for some N (theorem of whitney), so the distinction is not enormously important in theory, except of course when you wish to discuss a particular example that is not given as an embedded one.

this same situation arises in algebraic geometry where objects like the grassmanian of lines in three space can be given an abstract structure as an algebraic variety, by covering it with compatible affine open sets, and also can be somewhat cumbersomely embedded in projective space as a closed subvariety.

 

[mathwonk]

the wikipedia article

http://en.wikipedia.org/wiki/Manifold

looks very thorough and scholarly, but may not be written entirely to explain the subject clearly.

I.e. the writing style suggests that the author knows a great deal more than he is telling.
e.g. much is said but details are not always given precisely. e.g. whitney's theorem seems nowhere mentioned although the "extrinsic" and "intrinsic" viewpoints are explicitly contrasted.

does it help you?

 

[mathwonk]

here you go, a free chapter of the excellent looking book by Lee,

http://www.math.washington.edu/~lee/Books/smooth.html

 

[Mystic998]

Wow. Thanks for the speedy response.

And, yeah, I'm pretty sure it was mentioned as an aside in class that most interesting manifolds embed in R^n, it's just that the class started last week, and we aren't using that result yet. Hell, I don't even know what an embedding is beyond some intuition.

Also, Lee's book certainly looks more readable than Hirsch's Differential Topology and more pertinent than Guillemin and Pollack's book of the same name (they start out assuming all the manifolds are in R^n immediately).

 

[mathwonk]

guillemin and pollack is very smoothly written, i.e. very easy to read.

milnor is terrific. hirsch is an expert but writes poorly in my opinion.

 

[masnevets]

And, yeah, I'm pretty sure it was mentioned as an aside in class that most interesting manifolds embed in R^n, it's just that the class started last week, and we aren't using that result yet. Hell, I don't even know what an embedding is beyond some intuition.

Any smooth n-dimensional manifold (with a countable basis) can be embedded into 2n-dimensional Euclidean space. This is the Whitney embedding theorem. An embedding is the following:

Say you have a smooth map f: M -> N. Then f(M) is a subspace of N that has an induced topology (the subspace topology). If f: M -> f(M) is a homeomorphism, then we say f is an embedding. In the case of Euclidean space, it just means you can "put" the manifold in Euclidean space in such a way that its topology coincides with the subspace topology it would inherit.

For low n, say n=1,2, the theorem is tight. For example, the circle S^1 is a 1-manifold that cannot be embedded in R^1, and the Klein bottle is a 2-manifold that cannot be embedded in R^3.

See http://en.wikipedia.org/wiki/Whitney_embedding_theorem for more information.

 

[mathwonk]

Interestingly, it seems to be open whether the embedding can be chosen to have closed image, although this seems generally assumed to be true.