# How to prove the isotopy relative to a point exist?

Let $M$  be a differential manifold, and $f$ a diffeomorphism on $M$ which is isotopic to $id$. Assuming that $x\in M$ is a fixed point of $f$ and the orbit of $x$ under the isotopy is a trivial loop(trivial in $\pi_1(M)$). How to prove that there is an isotopy from $id$ to $f$ relative to $x$? (i.e. there is an isotopy $f_t$ satisfied that $f_0=id$, $f_1=f$ and $f_t(x)=x$.

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The quick argument is to use the fibration $Diff(M,x)\to Diff(M) \to M$ whose total space is the diffeomorphism group of $M$ and whose fiber is the subgroup fixing the point $x$. The projection $Diff(M)\to M$ is given by evaluation at $x$. Since this is a fibration, we have $\pi_1(Diff(M),Diff(M,x)) \cong \pi_1(M)$, this isomorphism being induced by the projection map of the fibration, evaluation at $x$. This isomorphism gives exactly what you want: An isotopy from the identity to a diffeomorphism fixing $x$ can be deformed to fix $x$ at all times, provided that the loop traced out by $x$ during the isotopy is trivial in $\pi_1(M)$.