Example of rationally acyclic open non-oriented manifold.

Question

A manifold is rationally acyclic if its reduced cohomology is zero over rational. If the manifold is closed odd dimensional or closed oriented even dimensional then it is not possible due to Euler characteristics and Poincare duality. In closed non-oriented case, we have very nice examples even-dimensional real projective spaces. It is possible in case of the open non-oriented manifold or not?

Answer 1

Sure. Let $M$ be a rationally acyclic manifold; then so is $M\times\mathbb{R}^n$ for every $n$.