Given a smooth compact $n$-dimensional manifold $M^{n}$, let $\operatorname{Diff}(M)$ denote the group of smooth diffeomorphisms $M \rightarrow M$ equipped with the Whitney $C^{\infty}$-topology. Let $h_{\phi} \colon [0, 1] \times M \rightarrow M$ denote the homotopy associated to a path $\phi \colon [0, 1] \rightarrow \operatorname{Diff}(M)$. My question is: Given such $\phi$, how can one prove that there exists a path $\psi \colon [0, 1] \rightarrow \operatorname{Diff}(M)$ with the same endpoints as $\phi$ and such that $h_{\psi}$ is smooth?

So far, I have neither succeeded in finding a formal proof nor a detailed reference in the literature. I am particularly interested in the case where $M$ has non-empty boundary. I am looking forward to helpful comments!

  • 1
    $\begingroup$ $Diff(M)$ is a Fréchet manifold. So you juqt have to solve the analoguous question in a Fréchet vector space. $\endgroup$ – Vincent H Mar 3 '16 at 22:00

Here is an example of a very hands-on (and standard) construction of the path that you're looking for. The goal will be take a sequence of diffeomorphisms in the path that are located sufficiently close to each other in the $C^\infty$ topology, and then to join these diffeomorphisms by smooth paths.

Choose an auxiliary Riemannian metric on $M$, and consider the "exponential map" $E \colon TM \to M \times M$ given by $v \mapsto (p(v),\exp_{p(v)}(v))$ where $p \colon TM \to M$ is the tangent bundle.

For a diffeomorphism $\varphi$ of $M$ we write $\Gamma_\varphi:=\{(x,\varphi(x))\} \subset M \times M$ for its graph (this is a smooth submanifold). Given that a diffeomorphism $\varphi$ is sufficiently $C^\infty$-close to the diagonal $\Gamma_{\mathrm{id}_M} :=\{(x,x); x \in M \} \subset M \times M$, it follows that $E^{-1}(\Gamma_\varphi) \subset TM$ consists of a smooth section $\zeta$ of $TM \to M$ together with possibly additional loci contained outside of some a priori fixed tubular neighbourhood of the zero-section. (Here I am omitting some applications of standard transversality results as well as properties of the exponential map.)

In other words $x \mapsto \exp_{x}(t\zeta)$ is the sought smooth path parametrised by $t$ which connects $\mathrm{id}_M$ (at $t=0$) and $\varphi$ (at $t=1$).

| cite | improve this answer | |

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.