🚀 go-pugleaf

pratt@Sunburn.Stanford.EDU (Vaughan R. Pratt) writes:

> smith@gramian.harvard.edu (Steven Smith) writes:
> >pratt@Sunburn.Stanford.EDU (Vaughan R. Pratt) writes:
> >
> >> What I don't understand here is how "living in R^n" is of itself
> >> creating complexity and obscurity.  I can see that the
> >> arbitrariness of f might get in the way.  But where does the
> >> complexity and obscurity creep in if for example we define a
> >> manifold to be a smooth retract of an open subset of R^n?
> >
> >The choice of ambient space is also somewhat arbitrary.  If the
> >manifold can be imbedded in R^n, then it can also be imbedded in
> >R^{n+1}, to say nothing of imbeddings in other manifolds.  The point
> >is that manifolds exists independently of their imbeddings in R^n or
> >other manifolds.  Any reasonable definition should possess this
> >invariance.
> 
> This won't wash, you've just replaced one arbitrariness with another
> more complicated one, namely the notion of atlas.

Did not! Did not!  We have been discussing locally Euclidean spaces.
Only if you would like to define a differentiable structure on a
locally Euclidean space is it necessary to introduce the notion of an
atlas.

> A MANIFOLD is a pair (M,F) consisting of a topological space M and an
> atlas F on M.

For argument's sake, lets call a locally Euclidean space a manifold,
and a locally Euclidean space endowed with a differentiable structure
(atlas) a differentiable manifold.

> Now why do you object to my arbitrariness and not the arbitrariness of
> atlases?  You chop up a manifold in some arbitrary way into overlapping
> pieces and then paste it back together with charts that coordinate
> smoothly at the overlaps.  For example to chart a 2-sphere you might
> chop it up into the northern and southern hemispheres, slightly
> extended to open sets so that they overlap at the equator, and then
> glue them back together "seamlessly".  But you might equally well chop
> it into east and west, or into 240 pieces, one per country or sea.
> What is nonarbitrary about this chopping?

Absolutely nothing.  No one ever said that arbitrariness is
necessarily bad; only that one should be honest about it.  Endowing a
manifold with an (arbitrary) atlas is a good honest way of defining
the notion of a differentiable function f: M -> R, which is otherwise
nonsensical.

> The only arbitrariness in my definition is the choice of neighborhood
> of the manifold and its retraction onto the manifold.  No chopping, no
> charts.

Sorry, but what is your definition specifically?  If manifolds are
simply retracts of open sets of Euclidean space, then using this
definition, how is projective space a manifold ?

> Furthermore my definition is a *lot* simpler, I'd say, even without
> going into the notion of smooth coordination required to define an
> atlas.
> 
> There has to be some other reason why the complicated notion of atlas
> is essential.

The atlas defines the notion of a differentiability.

A standard example illustrates this idea: Define the differentiable
manifolds M_1 = (R, A_1) and M_2 = (R, A_2) where A_1 contains the
homeomorphism t -> t and A_2 contains the homeomorphism t -> t^3.
Clearly, A_1 != A_2.  (However, M_1 and M_2 are diffeomorphic, but
there are examples of manifolds with nondiffeomorphic differentiable
structures.)

Steven Smith