Poincare dual of a point Why is the closed Poincare dual of a point in $R^n$ trivial but the compact  dual is a "bump"?
Please provide very detailed answer. 

Let M be an oriented manifold. Recall that the closed Poincare dual of a $k$-dimensional submanifold of $M$ is an element in $H^{n-k}(M)$. In our case, as a point is $0$-dimensional and $H^n(\mathbb{R}^n)=0$, there is nothing to compute.
The compact dual should be some $\omega\in H_c^n(\mathbb{R}^n)$, such that for every closed $0$-form $\tau$ one has $$\int_{\mathbb{R}^n}\omega\wedge\tau=\int_p\tau,$$where $p$ denotes our point. Since a closed $0$-form is in fact a constant function, this equality holds for $$\omega=fdx_1\wedge\ldots\wedge dx_n,$$where $f$ is a function with compact support whose integral is equal to $1$.
