Copy-pasting cycles on the double manifold makes cycle even?

Question

Let $M$ be an oriented compact manifold with non-empty boundary $\partial M$. Let $2M$ be the double of $M$ obtained by gluing two copies of $M$ along the boundary ($2M=M\cup_{ \partial M} \overline M$). Let $\eta\in H_{n-2}(M;\mathbb Z)$ be some ${n-2}-$cycle on $M$. It is well know that such classes can be represented by a submanifold, say $N\subset M$. Consider now the submanifold $\tilde N=N\cup \overline{N}\subset 2M$ obtained by considering two copies of $N$ in the double. This represents a homology class $\omega\in H_{n-2}(2W;\mathbb Z)$. Is $\omega=2u$ for some class $u\in H_{n-2}(2W;\mathbb Z)$?