De Rham interpretation of $H^1(R,p,\mathbb{C})$

Question

Let $R$ be a Riemann surface of genus $g\ge 2$ and $p\in R$ a point. This is my question:

Is there a way to interpret the relative cohomology group $H^1(R,p,\mathbb{C})$ as a De Rham cohomology group (i.e. as closed complex forms of some kind)? I thought it was the space of closed complex 1-forms with only one zero in $p$, but I can't prove it. Is it right?

Answer 1

No, it's not right. To do relative deRham cohomology, in your relative cochain complex $C^k(X,A)$ consists of $k$-forms on $X$ whose pullback to $A$ vanishes. This is a bit of a subtle game. But, in particular, when $A$ is a point, any $k$-form with $k\ge 1$ will pull back to be $0$. It's hard to find references for this, but I know of a book (in French) by Godbillon, Éléments de Topologie Algébrique that has all sorts of gems in it, including this material. There's also a more general discussion on pp. 78-79 of Bott and Tu's Differential Forms in Algebraic Topology.