I am currently reading an article https://arxiv.org/pdf/1802.02609.pdf, and a question arose.
The setting is at the beginning of section 2. Let $M$ be a closed, oriented, smooth $d$-manifold, and fix an embedded disck $D^d\subset M$ and a colloar neighborhood $[0,1]\times S^{d-1}\subset M\backslash \text{int}(D^d)$ such that $\{1\}\times S^{d-1}$ is identified with $\partial D^d$. Let $\gamma\in \Omega SO(d)$ and define a path $\lambda:I \to \text{Homeo}(M)$ by $$ \lambda(t)(m)=\begin{cases} \gamma(t)m & m\in D^d \\ (s,\gamma(s)x) & m=(s,x)\in [0,t]\times S^{d-1} \\ (s,\gamma(t)x) & m=(s,x)\in(t,0]\times S^{d+1} \\ m & \text{otherwise}\end{cases} $$ The autor says, that one can then use a suitable family of bump functions to make it into a path which takes values in $\text{Diff}^+(M)$ the topological group of orientation-preserving diffeomorphisms. Furthermore, this should be done such that the endpoints of $\lambda$ are preserved.
The idea i had was to use a standard partition of unity argument for a fixed covering of $[0,1]\times S^{d-1}$ as in Lee's book in the section for Whitney approximation theorem. Namely, one embeds $M\subset \mathbb{R}^n$ and gets a tubular neighborhood $U$. We can then use the POU, to make that path into a path $I\to C^\infty(M,U)$ and postcomposing with the projection $U\to M$ yields pointwise a smooth map $M\to U \to M$. However, doesn't this only yield a path $I\to C^\infty(M,M)$? Futhermore, this doesn't fix the endpoints. Any help or ideas would be appreciated.
Edit: As mentioned in the comments of the accepted answer, one can assume that $\gamma$ is a smooth loop for the purpose in the article. Moreover, i don't think it is possible to give a nice solution to the question as asked without this assumption if there even exists one.
It seems that a more straightforward approach is available; the interpolation doesn't require anything as involved as a partition of unity.
Choose a smooth function $\psi:\mathbb{R}\to\mathbb{R}$ satisfying $\psi(x)=0$ for $x<1/4$, $\psi(x)=1$ for $x>3/4$. We can then define the path $\lambda$ by $$ \lambda(t)(m)=\begin{cases} \gamma(t)m & m\in D^d \\ (s,\gamma(t\psi(s))x) & m=(s,x)\in[0,1]\times S^{d-1} \\ m & \text{otherwise} \end{cases} $$ This path only agrees with the original definition outside of the collar (necessarily, since the endpoints may fail to be smooth on the collar as originally defined), but it should still have all of the desired properties.