Rational homotopy type of the space of immersions of a manifold in an euclidian space

Let $M$ be a simply-connected $m$ dimensional manifold of finite type and $k$ a positif integer. In this paper we show that the rational Betti numbers of each component of the space of immersions of $M$ in ${\mathbb{R}}^{m+k}$, have polynomial growth. As consequence, we deduce that, if $M$ is a manifold with Euler characteristic $\chi \left(M\right)\le -2$, the Betti numbers of smooth embeddings, $Emb(M,{\mathbb{R}}^{m+k})$, have exponential growth if $k\ge m+1$. The main tool of this work is the construction of an explicit model of the space of immersions.