Topology of function spaces?

A standard reference for this is Hirsch's Differential Topology textbook. If $X$ is compact near all the topologies you'd like to consider are essentially the same. Sometimes they're called the Whitney topologies, or the $C^\infty$-topology, there are weak and strong variants that are only relevant when $X$ is non-compact.

These spaces have the homotopy-type of the space of continuous maps. The basic idea is to consider $Y$ as a submanifold of some Euclidean space, any continuous map $f : X \to Y$ you can apply a smoothing operator to, then project via the tubular neighbourhood theorem to get a smooth map $X \to Y$ approximating $f$ (in the $C^k$ sense for any $k$ as large as makes sense for your given map $f$), similarly you can apply this construction to families of functions.

Hirsch doesn't bother to get into the details of the Frechet manifold structure on these mapping spaces but it's available. Kriegl and Michor's "Convienient setting for global analysis" is a fairly comprehensive (if daunting) reference for this. But there are other references out there that provide a modest amount of details.

http://www.ams.org/publications/online-books/surv53-index

We have one of the founders of this subject online -- Richard Palais. Perhaps he will have some comments eventually.

edit: Back to your question. Spaces of continuous maps $C^0(X,Y)$ are a rather traditional thing to study in algebraic topology. It really depends on what kinds of questions you have about these spaces. For example, if $X$ and $Y$ are Eilenberg-Maclane spaces $C^0(X,Y)$ has much to do with plain old cohomology. If your spaces $X$ and $Y$ have nice cell decompositions, you can frequently get at aspects of the homotopy-type of $C^0(X,Y)$ via obstruction theory. But in general these spaces are pretty complicated.

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.

First off, you want your source space, $X$, to be compact (technically, sequentially compact will do). If you don't have that, $C(X,Y)$ need not be locally contractible so no hope of a manifold structure, infinite dimensional or otherwise. If $X$ is compact, all the arguments for loop spaces follow through, the only thing is that there's a bit more variety in model spaces - the models are spaces of sections of vector bundles over $X$ which, for $X = S^1$, is quite simple but for general $X$ is more variable. However, all these spaces behave like $C^\infty(X,\mathbb{R}^n)$ for all intents and purposes.

The topology on $C^\infty(X,Y)$ is referred to as the "compact-open" topology, but as I've imposed the restriction that $X$ be compact you could think of it as being a uniform-type of convergence. Basically, open sets are formed by insisting that all derivatives up to a certain finite order are bounded by a certain bound - the Riemannian structure helps with defining this (though it isn't necessary). However, I prefer to think of the smooth structure in terms of the exponential map which says that the smooth structure on $C^\infty(X,Y)$ is precisely that which makes a map $Z \to C^\infty(X,Y)$ smooth if and only if $Z \times X \to Y$ is smooth.

As a Riemannian manifold, $C^\infty(X,Y)$ can certainly be given a Riemannian structure. However, it's a weak structure not a strong one. I'm not sure about general $X$, but for $X = S^1$ then it's the best that it can be whilst still being weak.

Regarding homotopy types, the inclusion $C^\infty(X,Y) \to C^0(X,Y)$ is a homotopy equivalence. Ryan's answer contains the idea on how to do that.

However, you're more interested in references. Ryan's already mentioned the canonical reference, here's a few others:

• nLab

  • Harry's mentioned the page on loop spaces
  • http://ncatlab.org/nlab/show/differential+topology+of+mapping+spaces

• my papers (and other stuff)

  • The differential topology of loop spaces
  • How to Construct a Dirac Operator in Infinite Dimensions (contains details on types of Riemannian structure in infinite dimensions)
  • The Co-Riemannian Structure of Smooth Loop Spaces
  • Constructing Smooth Manifolds of Loop Spaces