TITLE: Homotopy of space of immersions, Smale-Hirsch theorem
QUESTION [10 upvotes]: If $M$ and $N$  are  manifolds with $\dim M< \dim N$, we denote by $Imm\left(M,N\right)$ the space of immersions of $M$ in $N$.
Let $M$ and $M'$ be a simply connected manifolds of dimensions $m>0$. It is true that if $M$ is homotopic to $M'$, then for $k\geq m$, the spaces $Imm\left(M,\mathbb{R}^{m+k}\right)$ and $Imm\left(M',\mathbb{R}^{m+k}\right)$ are homotopic? i.e
if  $k\geq m$, then $M\simeq M'\Rightarrow Imm\left(M,\mathbb{R}^{m+k}\right)\simeq Imm\left(M',\mathbb{R}^{m+k}\right)$?
Thanks
Abdoul

REPLY [2 votes]: Note that when $M$ and $M′$ are simply connected, I have a positive answer in the rational case. I.e  
If $k\geq m$ is an odd integer, then $M\simeq M'\Rightarrow Imm\left(M,\mathbb{R}^{m+k}\right)\simeq_{\mathbb{Q}}Imm\left(M',\mathbb{R}^{m+k}\right).$
Here $"\simeq_{\mathbb{Q}}"$ denotes the rational homotopy equivalence.