and
Given two topological spaces $X$, $Y$ one may ask for the rational homotopy type of their mapping space $Maps(X,Y)$. Under good conditions this is a nilpotent space of finite type and hence admits a Sullivan model from which its rationalized homotopy groups and its rational cohomology groups may be read off.
See at Sullivan model of free loop space.
Let $n \in \mathbb{N}$ be a natural number and $f\colon S^n \to S^n$ a continuous function from the n-sphere to itself. Then the connected component $Maps_f\big( S^n, S^n\big)$ of the mapping space which contains this map has the following rational homotopy type:
where $deg(f)$ is the degree of $f$.
Moreover, under the canonical morphism expressing the canonical action of the special orthogonal group $SO(n+1)$ on $S^n = S\big( \mathbb{R}^{n+1}\big)$ (regarded as the unit sphere in $(n+1)$-dimensional Cartesian space) we have that on ordinary homology
the generator in $\left\{ \array{ H_{2n+1}\big( SO(n+1), \mathbb{Q} \big) \simeq \mathbb{Q} &\vert& n\, \text{even} \\ H_{n}\big( SO(n+1), \mathbb{Q} \big) \simeq \mathbb{Q} &\vert& n\, \text{odd} } \right.$ maps to the fundamental class of the respective spheres in (1), all other generators mapping to zero.
(Møller-Raussen 85, Example 2.5, Cohen-Voronov 05, Lemma 5.3.5)
See at Sullivan model of a spherical fibration for more on this.
Jesper Møller, Martin Raussen, Rational Homotopy of Spaces of Maps Into Spheres and Complex Projective Spaces, Transactions of the American Mathematical Society Vol. 292, No. 2 (Dec., 1985), pp. 721-732 (jstor:2000242)
Ralph Cohen, Alexander Voronov, Notes on string topology (arXiv:math/0503625)
Urtzi Buijs, Yves Félix, Aniceto Murillo, $L_\infty$-rational homotopy of mapping spaces, published as $L_\infty$-models of based mapping spaces, J. Math. Soc. Japan Volume 63, Number 2 (2011), 503-524 (arXiv:1209.4756, doi:10.2969/jmsj/06320503)
A spectral sequence computing the rational homotopy of mapping spaces:
based on
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