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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 ofeach component of the space of immersions of $M$ in $\mathbb{R}^{m+k}$, havepolynomial growth. As consequence, we deduce that, if $M$ is a manifold withEuler characteristic $\chi\left(M\right)\leq -2$, the Betti numbers of smoothembeddings, $Emb\left(M ,\mathbb{R}^{m+k}\right)$, have exponential growth if$k\geq m+1$. The main tool of this work is the construction of an explicitmodel of the space of immersions.

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