Ganesh Ujwal Posted December 13, 2014 Share Posted December 13, 2014 (edited) 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? --- 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 Link to comment Share on other sites More sharing options...
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