The free smooth path space on a manifold

Question

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). However, an inspection of his method shows that it does not immediately adapt to the situation above.

Answer 1

Yes.

The technical details are in Yet More Smooth Mapping Spaces and Their Smoothly Local Properties, 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, particularly Proposition 5.1. This contains the germ of the idea.

2. Yet More Smooth Mapping Spaces and Their Smoothly Local Properties, 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 about piecewise-smooth maps.