Does every smooth surjective function have a smooth right inverse?

[Ganesh Ujwal]

If [latex]r:I\rightarrow J[/latex] is a smooth surjective function between perfect subspaces [latex]I[/latex] and [latex]J[/latex] of [latex]\mathbb{R}[/latex], can we always find a right inverse smooth function [latex]s : J \rightarrow I[/latex], i.e. [latex]r\circ s = id_{J}[/latex]?

In the same fashion, does every smooth injective [latex]s:I\rightarrow J[/latex] have an smooth injective left inverse?

 

A necessary condition is for the derivatives of [latex]r[/latex] and [latex]s[/latex] to be non-singular (in [latex]s(J)[/latex] and [latex]J[/latex] or in [latex]r(I)[/latex] and [latex]I[/latex] respectively at least).

So one should at least assume that.

This also implies that [latex]r[/latex] and [latex]s[/latex] are locally invertible there.

 

For example:

Loosening the question a bit,

if [latex]s : J \rightarrow I[/latex] is continuous and injective, then by the intermediate value theorem we can conlude that [latex]s[/latex] is monotone on every connected component of [latex]J[/latex].

If [latex]J = [a,b][/latex] is a compact interval, one can define a retraction [latex]r : I \rightarrow J[/latex] by inverting [latex]s[/latex] on [latex]s(J)[/latex] whilst being constantly [latex]a[/latex] or [latex]b[/latex] on the parts above and below [latex]s(J)[/latex] in [latex]I[/latex].

But what is if we really talk about smooth functions?

 

Where can I find a discussion on this and are there some nice counter-examples?

 

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Maybe it is fruitful to also generalize and rephrase this question in terms of categories.

 

I want to investigate the relations between the following kinds of maps in [latex]\mathcal{C}[/latex]:

 

- surjective maps [latex]o[/latex]

- injective maps [latex]i[/latex]

- right-cancellable maps (epics) [latex]e[/latex]

- left-cancellable maps (monics) [latex]m[/latex]

- right-invertible maps (split epics/retractions) [latex]r[/latex]

- left-invertible maps (split monics/sections) [latex]s[/latex]

 

where [latex]\mathcal{C}[/latex] is some adequate category of topological/smooth spaces.

 

In the square brackets stands the name I’d prefer to use for maps with the corresponding property.

Categorically we have the implications “[latex]r \Rightarrow e[/latex]” and “[latex]s \Rightarrow m[/latex]”.

In concrete categories I understand we also have “[latex]o \Rightarrow e[/latex]” and “[latex]i \Rightarrow m[/latex]”.

 

Now, I’m mainly interested in the implications “[latex]e \Rightarrow r[/latex]” and “[latex]m \Rightarrow s[/latex]“, that is:

> For which categories [latex]\mathcal{C}[/latex] of euclidian (topological/smooth) spaces is:

>

> - every epic a retraction, and

> - every monic a section?

 

And I’d be more than happy to have an answer only for categories in which the objects are perfect subspaces of [latex]\mathbb{R}[/latex] and morphisms are [latex]C^1[/latex] or [latex]C^\infty[/latex].

 

Edited December 13, 2014 by Ganesh Ujwal