Subject:
Re: question about smoothness
From:
Izak Grguric <grguric@math.ubc.ca>
Date:
Thu, 20 Jan 2005 14:12:15 -0800 (PST)
To:
Don Davis <dmd1@lehigh.edu>

For a neat proof of this see:
Kosinski: "Differential Manifolds"
page 48
Cor 2.6:
If f,g : M-> N are smooth and homotopic (as continuous maps),
then they are smoothly homotopic.

This Corollary comes as an immediate consequence of part (b) of the Theorem 2.5 preceding it. The proof of 2.5 uses: the existence of embeddings into R^n, existence of tubular neighborhoods, and then proceeds via a partition of unity argument. Nothing surprising there.

Hirsch: "Differential Topology" might contain something similar, but probably coated in the language of function spaces.

------


On Thu, 20 Jan 2005, Don Davis wrote:

> Subject: Question
> From: "boccellari" <boccellari@inwind.it>
> Date: Wed, 19 Jan 2005 20:29:05 +0100
>
>
> I would like to have a reference about the following problem
> whose answer should be well known.
>
> Consider two smooth maps f,g : S^n ------> S^m between spheres,
> suppose they are homotopic.
>
> Is it true that they are homotopic through a smooth map
> H : S^n x I -------> S^m ?
>
> Thank you for your kind attention
>
> Tommaso Boccellari
>