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 $M'$ 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

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