Can someone give me an example of a closed smooth oriented manifold $M$ and an orientation preserving diffeomorphism $f:M\rightarrow M$ such that $f^k$ is isotopic to the identity for some $k\geq 1$, but $f$ is not isotopic to any finite order diffeomorphism? Similarly for homeomorphisms, and also for homotopies.


Of course, Kerckhoff's famous theorem says that this is impossible in dimension $2$.



Counterexample to high dimensional Nielsen realization problem
This is discussed at great length (with copious references) in [Jonathan Block's and Shmuel Weinberger's paper.](http://math.uchicago.edu/~shmuel/Nielsen.pdf) The title of the paper is the suggestive "On the generalized Nielsen realization problem" (comm math helv, 2008)


