Why is it impossible for a compact connected non-orientable $n$-manifold to be the suspension of some connected based space?

Question

Suppose $M$ is a compact $n$-manifold (without boundary) that equals the reduced suspension $\Sigma Y$ of a connected based space $Y$. Why must $M$ be orientable?

I am aware that cup products vanish on suspensions and that cohomology is stable, and that $H_n(M;\mathbf Z)\cong\mathbf Z$ if $M$ is orientable, $0$ if M is not orientable.

Answer 1

The suspension of a (path-)connected space is simply connected (this is easy using van Kampen's theorem on the two cones that make up the suspension), and simply connected manifolds are orientable.