In this thesis we study the rational homotopy type of the space of immersions of a differential manifold M of dimension m in a differential manifold of dimension n, with n>m>0. This space is denoted Imm(M,N). If M and N are simply connected and of finite type, we prove that when all the Pontryagin classes of N are zero then each connected component of Imm(M,N) has the rational homotopy type of one connected component of some mapping space.In the particular case when N=R^n, we prove that each connected component of Imm(M,R^n)has the rational homotopy type of the product of the Eilenberg-Mac Lane spaces. We give an explicit description of each connected component and we prove that it depends only on the codimension n-m and the Betti numbers of M. If n>2m and if the Euler characteristic of M is less or equal to -2, then the Betti numbers of a space of embeddings of M in R^n have exponential growth. When M is without boundary we obtain in certain case an explicit formulas of the series of the ranks of homotopy groups of the connected component of Imm(M,R^n).

Homotopie rationnelle de l'espace des immersions entre variétés

In this paper we study the rational homotopy of the space of immersions, $Imm\left(M,N\right)$, of a manifold $M$ of dimension $m\geq 0$ into a manifold $N$ of dimension $m+k$, with $k\geq 2$. In the special case when $N=\mathbb{R}^{m+k}$ and $k$ is odd we prove that each connected component of $Imm\left(M,\mathbb{R}^{m+k}\right)$ has the rational homotopy type of product of Eilenberg Mac Lane space. We give an explicit description of each connected component and prove that it only depends on $m$, $k$ and the rational Betti numbers of $M$. For a more general manifold $N$, we prove that the path connected of $Imm\left(M,N\right)$ has the rational homotopy type of some component of an explicit mapping space when some Pontryagin classes vanishes.

Rational homotopy of the space of immersions between manifolds

Type d'homotopie rationnelle relative des fibrés en variétés de Stiefel

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Type d'homotopie rationnelle relative des fibrés en variétés de Stiefel