Let $X$ be an open manifold, with one end $N$,
Q How to show that $H^1_{c,dR}(X)\to H^1_{dR}(X)$ is an injective,? here $H^1_{dR}$ denotes the de Rham cohomology and $H^1_{c,dR}$ denotes the compactly supported de Rham cohomology.
Suppose that $H^1(N;\mathbb R)=0$.
Q How to show that $H^1_{c,dR}(X)\cong H^1_{dR}(X)$?