When compactly supported cohomology ring is zero?

Question

The compactly supported cohomology (over rational) of a compact manifold is the same as ordinary cohomology. Also, we can relate the compactly supported cohomology of the oriented non-compact manifold with ordinary cohomology using Poincare duality. How we can understand the compactly supported cohomology of the open non-oriented manifold. For example, compactly supported cohomology of open Mobius band is zero. My first question is that when compactly supported cohomology is zero? My Second question is that if a manifold is open non-oriented and M is rationally acyclic then it compactly supported cohomology is zero or not? In case of compact or open oriented manifold, the compactly supported is not zero due to Poincare duality.