Rational homotopy of the space of immersions between manifolds

TL;DR

This paper investigates the rational homotopy types of the space of immersions between manifolds, providing explicit descriptions and conditions under which these spaces have simplified rational homotopy structures.

Contribution

It offers a detailed analysis of the rational homotopy of immersion spaces, especially for Euclidean targets and under Pontryagin class vanishing conditions, with explicit descriptions.

Findings

Connected components of immersion spaces in Euclidean space have rational homotopy type of products of Eilenberg-Mac Lane spaces.

The rational homotopy type depends on the dimensions and Betti numbers of the source manifold.

For general manifolds, the homotopy type relates to explicit mapping spaces when Pontryagin classes vanish.

Abstract

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.