Suppose $(E, M, π)$ is a vector bundle. I want to show that projection map
$π: E\longrightarrow M$ is a homotopy equivalence. I know for every open set $U$ in $M$ we have a diffeomorphism (local trivialization) $$φ \colonπ^{-1} (U) \longrightarrow M×R^n .$$ I know that every diffeomorphism is a homeomorphism and so is a homotopy equivalence. But I don't know how to show $π: E\longrightarrow M$ is a homotopy equivalence.
Let $s$ be the zero section, we have $\pi\circ s= Id_M$.
Let $x\in M$, $v\in E_x$, the map $\pi_t(v)=tv$ is well defined $\pi_1=Id_M$,$s\circ \pi=\pi_0$. This is equivalent to saying that $s\circ \pi$ is homotopic to $Id_M$.