Let $M$ be a closed, smooth manifold and let $PM$ be the space of unbased piecewise smooth paths $[0,1] \to M$. Then restricting a path to its boundary gives a map
$$
PM \to M \times M .
$$


**Question** is this map a fiber bundle?


Andrew Stacey showed that a related map, the free smooth loop fibration $LM \to M$, is a fiber bundle (see *The differential topology of loop spaces*, arXiv:[math/0510097](https://arxiv.org/abs/math/0510097)). However, an inspection of his method shows that it does not immediately adapt to the situation above. 



The free smooth path space on a manifold
Yes.


The technical details are in *[Yet More Smooth Mapping Spaces and Their Smoothly Local Properties](https://arxiv.org/abs/1301.5493)*, specifically in Section 5 which establishes that smooth manifolds are *smoothly locally deformable* which means that there are lots of diffeomorphisms flying around.
Interestingly, although I considered subspaces I didn't consider spaces over other spaces. Nonetheless, the same technology allows us to do so.


Let $M$ be a smooth manifold. Section 5 of *Yet More ...* shows that $M$ is *smoothly locally deformable*. In the discussion preceding Proposition 3.12 it is shown that this means that there is a neighbourhood $M \subseteq V \subseteq M \times M$ and a smooth map $\phi \colon \mathbb{R} \times V \to \operatorname{Diff}(M)$ with the following properties:


1. For $v \in V$, $t \mapsto \phi\_{t,v}$ is a group homomorphism $(\mathbb{R},+) \to \operatorname{Diff}(M)$.
2. For $t \in \mathbb{R}$ and $v = (x,y) \in V$, $\phi\_{t,v}$ is the identity outside $V\_x := \{x' : (x,x') \in V\}$.
3. For $v = (x,y) \in V$, $\phi\_{1,v}(y) = x$.


Now let $T$ be a compact smooth space and $S \subseteq T$ a compact subset. We assume that there is a neighbourhood $S \subseteq U \subseteq T$ with a retraction $\tau \colon U \to S$, and a bump function $\sigma \colon T \to [0,1]$ such that $\sigma(S) \subseteq \{1\}$ and $\overline{\sigma^{-1} (0,1]} \subseteq U$.


Fix a class of function that is closed under diffeomorphism and which satisfies a sheaf condition in that functions can be defined locally.


Let $\alpha \colon S \to M$ be a function. Define $C\big((S,T),(V,M)\big)\_\alpha$ to be the space of functions $\beta \colon T \to M$ with the property that $(\alpha, \beta\mid\_S)$ maps $S$ into $V$. Define $C(T,M)\_\alpha$ to be the space of functions $\beta \colon T \to M$ such that $\beta\mid\_S = \alpha$. Define $C(S,V)\_\alpha$ to be the space of functions $\beta \colon S \to M$ such that $(\alpha,\beta)$ maps $S$ into $V$ (I'm not sure my notation is the best here!).


We define $\Phi \colon C\big((S,T), (V,M)\big)\_\alpha \to C(T,M)\_\alpha \times C(S,V)\_\alpha$ as follows. The map to the second factor is simply the restriction to $S$. The map to the first factor takes a function $\beta \colon T \to M$ to the function:


$$
t \mapsto \begin{cases}
\phi\_{\sigma(t), (\alpha(\tau(t)), \beta(\tau(t)))}\big(\beta(t)\big) & t \in U \\\\
\beta(t) & t \notin U
\end{cases}
$$


The conditions on $\phi$ mean that this patches together to give a well-defined function. The inverse of $\Phi$ takes a pair $(\beta,\gamma)$ to:


$$
t \mapsto \begin{cases}
\phi\_{-\sigma(t), (\alpha(\tau(t)),\gamma(\tau(t)))}\big(\beta(t)\big) & t \in U \\\\
\beta(t) & t \notin U
\end{cases}
$$


The case in point uses piecewise-smooth functions, $T = [0,1]$ and $S = \{0,1\}$. The conditions are easily checked.


**Further Reading**


1. *[The differential topology of loop spaces](https://arxiv.org/abs/math/0510097)*, particularly Proposition 5.1. This contains the germ of the idea.
2. *[Yet More Smooth Mapping Spaces and Their Smoothly Local Properties](https://arxiv.org/abs/1301.5493)*, this contains the technical results needed. Proposition 3.12 is quite close to what you need here. This would establish that $LM \subseteq PM$ has a tubular neighbourhood, which says that it is a bundle on a neighbourhood of a diagonal. Interestingly, I didn't consider fibrations of mapping spaces. Maybe I should add another section ...
3. *[The Smooth Structure of the Space of Piecewise-Smooth Loops](https://arxiv.org/abs/0803.0611)* about piecewise-smooth maps.


