0
$\begingroup$

In Guillemin and Pollack's Differential Topology, they (roughly speaking) define a manifold to be a space which is locally diffeomorphic to Euclidean space. Now this is obviously not the full definition they give, however, the gist of my question does not depend on the other details.

My question is, did Guillmin & Pollack give a definition of a smooth manifold instead of an arbitrary manifold (excluding the fact that the already assumed the manifold to be embedded in some ambient euclidean space)? I was under the impression that a manifold was defined to be locally homeomorpic to Euclidean space and that the property of a smooth mapping between the manifold and euclidean space would define a smooth manifold.

This would obviously make sense since the book is a differential topology text and not necessarily interested in arbitrary manifolds.

Also, the definitions of a smooth manifold that I have seen previously were done in terms of charts, atlases, and the requirement that the transition functions be smooth functions. Is the definition given by G & P comparable to the ones I am more familiar with?

$\endgroup$
1
  • 3
    $\begingroup$ Yes their definition excludes non-smoothable topological manifolds because they implement smoothness from the onset as you noticed. The term manifold is context-dependent and for a differential topologist the attribute "smooth" is usually understood. Anyway, by the Whitney embedding theorem no smooth manifold is left out by their definition so yes, theirs is perfectly equivalent to the ones you already know (provided the abstract definitions include Hausdorffness and second countability, which hold automatically in the Guillemin&Pollack's definition because of the embedding). $\endgroup$
    – johndoe
    Jan 22, 2016 at 1:44

1 Answer 1

3
$\begingroup$

This is just an expansion of johndoe's comment.

The normal definition of differential manifold requires charts to be able to define the concept of "diffeomorphic". By making their manifolds come with an embedding, they can borrow the concept from $\Bbb R^n$ instead. However, if one forms a collection of these local diffeomorphism, it will satify all the conditions of an atlas. Thus this definition implies the normal one.

And since any manifold by the normal definition is embeddable in $\Bbb R^n$ for some $n$, the normal definition also implies this one, up to diffeomorphism.

One shortcoming of this approach is that it makes it harder to see what are properties of the manifold itself vs properties of the embedding. Indeed, I believe this was the main impetus behind the common definition of manifolds as abstract objects. Another shortcoming is that it is less clear how the concept can be extended, for example, paracompact manifolds.

$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .