Retraction, inclusion and homotopy

Question

Let $M$ be a manifold, $S$ a submanifold of $M$, $r : M\rightarrow S$ the retraction map and $i : S \rightarrow M$ the inclusion map.

I know that the restriction $r_{|S}$ is equal to $id_S$, the identity map of $S$ in $S$.

Can I say that $r\circ i \sim_{\infty} id_S$, where $\sim_{\infty}$ is the homotopical equivalence?

I thought of this because $r\circ i$ can work as the restriction of $r$ over $S$ and the fact that $r_{|S} = id_S$ implies that $r_{|S} \sim_{\infty} id_S$ with a trivial homotopy.

Answer 1

Of course. You have $r \circ i = id_S$ which is much stronger than $\sim_\infty$.