Let $X$ and $Y$ be manifolds and suppose that $X$ is a compact, complete metric space and $Y$ is a complete metric space. So, both $X$ and $Y$ are Baire spaces.
Question: For what values of $k\geq 0$ is the space $C^k(X,Y)$ (with the $C^k$-topology) a Baire space?
For $k=0$, the space $C^0(X,Y)$ is a Baire space because it is a complete metric space with respect to the uniform metric (since $Y$ is complete).
What about the case $k=\infty$?
References are also very much appreciated.
The short answer is: yes, they are Baire spaces for all $k$, including $k = \infty$.
As long as $X$ is compact, there is essentially no choice about the "correct" topology on $C^k(X,Y)$ and the $C^0$ case elegantly implies the $C^k$-case using the device of $k$-jet bundles $J^k(X,Y)$. These yield a natural identification of $C^k(X,Y)$ with a closed subspace of $C^0(X,J^k(X,Y))$ (also for $k = \infty$).
If we drop compactness of $X$ (but assume that $X$ is finite-dimensional, hence locally compact), the same technique works, but now there are two distinct reasonable topologies on $C^r(X,Y)$, the weak and strong Whitney topologies. Those still have the Baire property (the strong topology is usually not metrizable).
A solid reference with full details is Hirsch, Differential Topology, chapter 2, especially section 4. See in particular Theorem 4.4 and the paragraphs following it.