Subject:
Re: Jeff Smith's email
From:
Nils Barth <nbarth@math.uchicago.edu>
Date:
Wed, 26 Jan 2005 15:04:28 -0600
To:
Don Davis <dmd1@lehigh.edu>

Dear Jeff (and mailing list readers),

I'm not disputing that these results are true, standard and easy;
however, the argument you outline confused me and others.
You state two "therefores" that are either false or require elaboration.

Let me explain my confusion, and a possible resolution.

Your claim is:
Smooth(M,N) < Cont(M,N)
  is a weak homotopy equivalence.
(where Smooth, Cont mean the spaces of smooth/continuous
 maps, respectively, and < means subset)

Here's how I read your argument:
- Smooth(M,N) < Cont(M,N) is dense
- and so every path in Cont(M,N) can be approximated by a path
  in Smooth(M,N)

It's the "and so" that I object to:
just because X < Y is dense does not mean that every path in
Y can be approximated by a path in X.

Now perhaps what you meant is:
- a continuous path in Cont(M,N) is the same as an element
  of Cont(I x M, N), via the adjunction
  Hom(X,Hom(M,N)) = Hom(X x M, N)
  ...and likewise for smooth paths
- I x M is also a manifold!
- Smooth(I x M, N) < Cont(I x M, N) is dense
- thus every continous path in Cont(M,N) can be approximated
  by a smooth path in Smooth(M,N)

This is still not quite enough:
we've replaced a continuous homotopy between two smooth functions f,g
by a nearby smooth homotopy between two NEARBY functions.

We need a relative statement:
let X, Y be manifolds, K < X a closed subset.
Given a continuous map H: X -> Y that is smooth on K,
there is a nearby smooth function H~: X -> Y that agrees
with H on K.

[This is Kosinski's 2.5(b);
 note that by "nearby" he doesn't just use one epsilon,
 but a functional on Y (since he doesn't assume compact).]

The result we want follows by letting
X = I x M
Y = N
K = {0} x M union {1} x M
H restricted to K = two smooth functions f, g
H = the continuous homotopy between f, g

and then H~ is the nearby smooth homotopy between f and g.
[This is Kosinski's Corollary 2.6;
 note that he doesn't give a proof of the Corollary, and so
 it was filling in the details that I understood (I presume)
 our confusion.]

     *     *     *     *     *     *     *

So perhaps the confusion was that you meant:
smooth maps are dense in continuous maps
in particular, Smooth(I x M, N) is dense in Cont(I x M, N)

...and I didn't fill this in in my head,
and you didn't think it bore elaboration.

     *     *     *     *     *     *     *

On the other statement, that density (and density of paths)
implies homotopy equivalence, is wrong without
added conditions:
Consider X = {1/n : n in N} and Y = X union {0}
Then
- X is dense in Y
- for all n,
  Hom(I^n, X) = X, Hom(I^n, Y) = Y
  [as X and Y are both totally disconnected]
  so Hom(I^n, X) is dense in Hom(I^n, Y)
- but X -> Y is not a weak homotopy equivalence:
  it's not even onto pi_0

Admittedly this is a gross point-set example,
but I don't know enough about the point-set topology of
function spaces to reject this out of hand.

     *     *     *     *     *     *     *

Hopefully this explains my concerns; if I'm missing
something and you've the time to elaborate, I'd be
appreciative.

best,
  nils


>> Subject: Re: 3 postings
>> From: Jeffrey H Smith <jhs@math.purdue.edu>
>> Date: Tue, 25 Jan 2005 11:34:47 -0500 (EST)
>> 
>> JFC
>> 
>> These are standard results from the fifties and sixties. The space of
>> smooth maps is dense in the space of continuous functions and so the every
>> continuous family of maps can be approximated within epsilon by a smooth
>> family of maps. From which it follows that the space of smooth maps is
>> weak equivalent to the space of continuous maps.