Let $X,Y$ be finite-dimensional differentiable manifolds, and let's assume that they are connected. In fact, in applications I would like both $X$ and $Y$ to be riemannian manifolds.
Let $C^\infty(X,Y)$ denote the space of smooth maps $f: X \to Y$. I'm interested in, say, the connected components, fundamental group,… of this space, but I'm really not sure where to start looking. I realise that I first need to topologise this space. My only experience in this realm is an introductory point-set topology course (based on Munkres) I took as a graduate student. Munkres talks about the compact-open topology for the space $C^0(X,Y)$ of continuous maps between two topological spaces and shows, for instance, that if $X$ is locally compact Hausdorff then the evaluation map $X \times C^0(X,Y) \to Y$ is continuous. Later in the book he also applies this to give a slick proof of the existence of covering spaces with prescribed covering group.
Back to the differentiable category, ideally I'd like to be able to do calculus on $C^\infty(X,Y)$, hence I'd like to think of $C^\infty(X,Y)$ as an infinite-dimensional differentiable manifold and possibly even riemannian whenever so are $X$ and $Y$.
In case it helps to focus the question, let me say a few words of (physical, I fear) motivation.
When $X,Y$ are riemannian, $C^\infty(X,Y)$ plays the rôle of the configuration space for a physical model known as the nonlinear sigma model, whose action functional, assigning to $\sigma: X \to Y$, the value of the integral (either take $X$ to be compact or else restrict the possible functions further to assure convergence)
$$ S[\sigma] = \int_X |d\sigma|^2 \operatorname{dvol}_X,$$
where I'd like to think of $d\sigma$ as a one-form on $X$ with values in the pullback $\sigma^*TY$ by $\sigma$ of the tangent bundle to $Y$, and $|d\sigma|^2$ involves the metric on the bundle $T^*X \otimes \sigma^*TY$ induced from the riemannian metrics on $X$ and on $Y$. The extrema of $S$ are then the harmonic maps.
We are often interested in the quantum theory (un)defined formally by a path integral. A mathematically conservative point of view is that the path integral simply gives a recipe for the perturbative treatment of the quantum theory, where we fix an extremum $\sigma_0$ of $S$ and quantise the fluctuations around $\sigma_0$. By definition, fluctuations around $\sigma_0$ lie in the connected component of $\sigma_0$ and as a first approximation, the path integral becomes a sum over the connected components of the space of maps. Hence the interest in determining the connected components of $C^\infty(X,Y)$, which in this context are often called superselection sectors.
So in summary, a possible question would be this:
What can be said about the topology (e.g., homotopy type) of $C^\infty(X,Y)$ in terms of $X$ and $Y$?
I'm not asking for a tutorial, just for some orientation to the available literature.
Thanks in advance.
Added
In response to the helpful answers I have received already, I'd like to point out that my interest is really on the homotopy type of the space of maps. I only mentioned the analytic aspects in case that narrows down the topology one would put on the space. As pointed out in the answers, most reasonable topologies are equivalent, so this is a relief.
As for concrete examples, I am particularly interested in the case where $X$ is a compact Riemann surface and $Y$ a compact Lie group. I'm happy to put the constant curvature metric on $X$ and a bi-invariant metric on $Y$.
Best Answer
Homotopy theory of function spaces is a healthy subfield of homotopy theory. See e.g. recent Oberwolfach report from a meeting on the subject.
If you share what $X,Y$ you are interested in, I may be able to say more, even though I am by no means an expert.
EDIT: In response to your edit, suppose $\Sigma$ is a compact Riemann surface, $G$ is a compact Lie group, and let's study $k$th homotopy group of $C^0(\Sigma, G)$. Actually, most of the discussion below applies to general $X,Y$ that are, say, compact manifolds or even finite CW-complexes, but let's stick to the case of interest. There is an well-known homeomorphism $C^0(S^k,C^0(\Sigma, G))\cong C^0(S^k\times\Sigma, G)$ given by the adjoint, and since any homeomorphism preserves path-components, we can identify $\pi_k(C^0(\Sigma, G))$ with $[S^k\times \Sigma, G]$, the set of homotopy classes of maps from $S^k\times \Sigma$ to $G$. Smashing $S^k\vee\Sigma$ inside $S^k\times\Sigma$ to a point gives $S^k\wedge\Sigma$, which is the $k$-fold suspension $S^k\Sigma$ of $\Sigma$. The cofiber sequence of the quotient map $S^k\times\Sigma\to S^k\wedge\Sigma$ is an exact sequence, namely, $$[S^k\Sigma, G]\to [S^k\times\Sigma, G]\to [S^k\vee \Sigma, G]=\pi_k(G)+[\Sigma, G].$$ Let's suppose that $G$ is simply-connected, so that $[\Sigma, G]$ is a point (as any Lie group has trivial $\pi_2(G)$ and $\Sigma$ is $2$-dimensional). Then the above cofiber sequence is an exact sequence of groups (not just sets). In trying to compute $[S^k\Sigma, G]$ it helps to recall that $G$ is rationally homotopy equivalent to the product of odd-dimensional spheres, so rationally $[S^k\Sigma, G]$ is the product of $[S^k\Sigma, S^m]$'s where $m$ is odd, and of course $[S^k\Sigma, S^m]=[\Sigma, \Omega^kS^m]$. This should allow you to do some computations.
Finally, I wish to comment that the inclusion $C^\infty(X,Y)\to C^0(X,Y)$ is a weak homotopy equivalence i.e. it induces an isomorphism of homotopy groups as any map of a sphere/disk is homotopic to a nearby smooth map. A result of Milnor says that $C^0(X,Y)$ is homotopy equivalent to a CW-complex, provided $X$ is a finite CW-complex (if memory serves me). I am not sure why the same is true for $C^\infty(X,Y)$ so at the moment I do not know if the above inclusion is a homotopy equivalence.