Consider the following claim: 



> 
> Let $p:M \to N$ be a (surjective) submersion of finite-dimensional smooth
>  manifolds. Let $J$ denote one of $[0,1],\ [0,1),\ (0,1]$. Then $p\_\*:M^J \to
> N^J$ is a submersion of Frechet manifolds, where $X^J$ denotes the usual manifold of smooth paths in $X$.
> 
> 
> 


The intuition is that given a lift of a path $\gamma$ through $p$, one can find
a neighbourhood $U$ of the image of $\gamma$ and a neighbourhood $V$ of the image of
the lift such that for every path in $U$ one can smoothly choose a lift to $V$.


Here I suppose we need a submersion of Frechet manifolds to be a map that
admits local sections through every point in the domain, if that is
the 'correct'
notion of submersion in that setting (certainly not the 'surjective on tangent
spaces' version).


I think the proof would use the characterisation of submersions as maps
which look locally (on both the *domain* and codomain) like projections
$U \times \mathbb{R}^n \to U$, and the existence of good open covers with smooth
contractions.


I think I'm able to prove that there are continuous sections through every point
in the domain, thinking of everything as a topological space, and using the
compact-open topology on the mapping spaces. But I don't know off the top
of my head that the compact-open topology on the space of smooth paths
is the same as the topology inherited from the Frechet manifold structure.
(My guess is that it is.)


My question: is the claim true?




---


As Andrew Stacey points out in the comments, the mapping space is not a manifold for non-compact intervals. However, I think I really only need maps which have all derivatives uniformly bounded (but a different bound for each derivative!). Since the topology on the mapping space for compact intervals uses uniform convergence, I'm betting that this set has the structure of a Frechet manifold.


Question 2: am I right?


Question 1': if so, is the claim true for this (putative) map of Frechet manifolds?



Induced map on path manifolds: is it a submersion?
This is answered affirmatively in [Yet More Smooth Mapping Spaces and Their Smoothly Local Properties](http://arxiv.org/abs/1301.5493). Specifically:


### Theorem 1.1


Let $M$ be a finite dimensional smooth manifold.
Let $S$ be a Frölicher space with the property that there is a non-zero smooth function $C^\infty(S,\mathbb{R}) → \mathbb{R}$ with support in $C^\infty(S,(-1,1))$.
Then $C^\infty(S,M)$ is a smooth manifold which is locally modelled on its kinematic tangent spaces.


Suppose, in addition, that $N$ is another finite dimensional smooth manifold and $f \colon M \to N$ a regular smooth map.
Then $C^\infty(S,f) \colon C^\infty(S,M) \to C^\infty(S,N)$ is a regular smooth map.


