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

TL;DR

This paper investigates the rational homotopy type of the space of immersions of a simply-connected manifold into Euclidean space, showing polynomial growth of Betti numbers and exponential growth under certain conditions, using explicit models.

Contribution

It constructs an explicit model of the space of immersions and analyzes the growth of Betti numbers, revealing new insights into their asymptotic behavior.

Findings

Betti numbers of immersion spaces grow polynomially

Betti numbers of embedding spaces grow exponentially when certain conditions are met

Explicit models of immersion spaces facilitate homotopy type analysis

Abstract

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 $\chi\left(M\right)\leq -2$, the Betti numbers of smooth embeddings, $Emb\left(M ,\mathbb{R}^{m+k}\right)$, have exponential growth if $k\geq m+1$. The main tool of this work is the construction of an explicit model of the space of immersions.