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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.

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  • $\begingroup$ I think Andrew told me this was true, but I can't recall where or how (I can't find an email about it). As far as the reference goes, it looks somewhat implicit: on page 4 Andrew says how section 5 proves $ev\colon LM \to M$ is locally trivial, as a consequence of general results, but section 5 is not forthcoming (on a quick scan) as to how. $\endgroup$ – David Roberts Nov 3 at 5:04
  • $\begingroup$ I'd go so far as to tentatively claim that $PM \to M \times M$ is probably locally homotopy trivial, at least in the case where $PM$ is the space of smooth maps, but I'd have to think a bit more as to why. $\endgroup$ – David Roberts Nov 3 at 5:07
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    $\begingroup$ Hi John. Firstly, feel free to email me directly: loopspace-usual symbol-mathforge-next usual symbol-org. Secondly, piecewise-smooth is problematic: see my article in Glasgow Math, also on arxiv, on that. With smooth, then PM -> MxM is locally trivial for the same reasons as 5.1, but the fibre is not LM. This is also covered in "Yet more smooth mapping spaces .." Prop 3.12. If you clarify what form of smooth mapping space you're happy with, I'll try to tidy that up into an answer. $\endgroup$ – Loop Space Nov 3 at 9:42
  • $\begingroup$ I forget which article is in Glasgow Math and it might not be the one I meant. All my relevant articles are on the arXiv. $\endgroup$ – Loop Space Nov 3 at 10:03
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    $\begingroup$ @JohnKlein The question is as to what you mean by smooth on a closed interval. Do the derivatives exist at the end points? Anyway, I've answered (I think!) your actual question. I'd love to hear more, but email might be better than comments here. $\endgroup$ – Loop Space Nov 3 at 22:18
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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.

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    $\begingroup$ Yes, it sounds as though you're less interested in the space in and of itself and more in relation to other, better behaved, spaces. Some uses of piecewise-smooth require it to be a manifold in and of itself and that's what my work focusses on, and I'm aware that that structure is often glossed over which is why I tend to get a bit emphatic when talking about piecewise-smooth stuff! My apologies if my tone was a bit over the top with that. $\endgroup$ – Loop Space Nov 3 at 23:00
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    $\begingroup$ Okay, I can fix this. It'll have to wait until later, but the idea is to pick different diffeos of M for $\gamma(1)$ and $\gamma(0)$ and then apply these diffeos at the different ends of the path. $\endgroup$ – Loop Space Nov 4 at 7:12
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    $\begingroup$ @LoopSpace Thanks. It looks right to me now. $\endgroup$ – John Klein Nov 5 at 3:41
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    $\begingroup$ @JohnKlein I've updated it again to generalise it. I'm considering adding this version to my paper. $\endgroup$ – Loop Space Nov 8 at 6:45
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    $\begingroup$ That's great! I can cite you then. $\endgroup$ – John Klein Nov 10 at 4:44

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