This is probably a fairly basic question:
Poincaré Duality states the following: Given an $n$-manifold, the $k^{th}$ homology is isomorphic to the $(n-k)^{th}$ cohomology.
So I was curious is there some certain relation you get when dealing with de Rham cohomology? For example lets take the punctured plane, then the 1st cohomology group is simply $\mathbb{R}$. However, when looking at the first homotopy group of the punctured plane, we get $\mathbb{Z}$.
But, these two groups are not isomorphic. What am I missing?