Let $M$ be an oriented manifold, not necessarily compact. Let $M'$ be a (finite) $k$-sheeted cover and let $\pi:M'\longrightarrow M$ be the covering map.

**Question 1** : *Is it true that $M'$ is (oriented) cobordant to $k$ disjoint copies of $M$?*

**Question 2** : *If the answer to the above is true then let $W$ be such a cobordism. Is there a map $\widetilde{\pi}:W\longrightarrow M$ such that $\widetilde{\pi}$ restricts to the natural maps on each end?*

I would guess that the answers to these questions should be well-known. However, any references, proofs or counter-examples(?) would be helpful.

EDIT : From the discussion/answers below, it is clear that the answer to (1) is positive when $M$ is a closed, oriented manifold. The characteristic numbers determine the oriented cobordism class of $M$. However, when $M$ is an *open manifold*, these numbers do not make immediate sense. I'm particularly interested in both questions in this scenario.