Homology of a region of the plane

Question

This is related to this MO question, I don't know if it's really "research-level". As in that question, let $U$ be a domain of the complex plane $\mathbb{C}$, i.e. an open connected subset. Let $$ \kappa_a:=\frac{1}{\zeta-a}d\zeta $$ be the Cauchy kernel centered at $a\in \mathbb{C}$. Of course, if $a \in \mathbb{C}-U$, $\kappa_a$ is a holomorphic 1-form on $U$. Given a continuous closed path $\gamma:[0,1]\to U$, $\gamma(1)=\gamma(0)$, we say that $\gamma$ is equivalent to $0$ in $U$ if for every holomorphic function $f\in\mathcal{O}(U)$, one has $$ \langle f\cdot\kappa_a,\gamma\rangle := \int_\gamma f\cdot \kappa_a = 0 $$ for every $a\notin U$.

1) Is it true that $\gamma$ is equivalent to $0$ in $U$ if and only if $[\gamma]=0\in H_1(U,\mathbb{Z})$?

A simple question on the homology of $U$:

2) Is it true that $H_1(U,\mathbb{Z})$ is the free module generated by $[\gamma_i]$, for $\gamma_i$, $i\in\pi_0(\mathbb{C}-U)$, loops each of which goes around each bounded connected component $S_i$ of the complement of $U$ exactly once, i.e. $\langle\kappa_{a_j},\gamma_i\rangle=\delta_{ij}$ for any choice of $a_j\in S_j$? [Edit: not at all sure it may be true; thinking of when $U$ is the complement of a Cantor set... But perhaps a weakened but still meaningful assertion holds?]

By Poincaré duality for non-compact manifolds, the homology (with complex coefficients) $H_i(U,\mathbb{C})$ is dual to de Rham cohomology with compact supports $H^i_{dR,\; c}(U,\mathbb{C})$. (By the way, the latter is isomorphic to the cohomology of the complex $\Gamma(U,\Omega^{\bullet})$ of holomorphic differential forms. See the other question.)

3) Is it possible to find 1-forms $\alpha_i$ on $U$, $i\in\pi_0(\mathbb{C}-U)$, with compact support, such that $\alpha_i$ is cohomologous to $\kappa_{a_i}$ and the $[\alpha_i]$ generate over $\mathbb{C}$ the cohomology $H^1_{dR,\; c}(U,\mathbb{C})$? (I'd say no.) Anyway, how to see question (1) in the light of Poincaré duality and/or the remarks above on cohomology?

Answer 1

Integrating the form $\kappa_a$ computes the winding number of $\gamma$ about $a$. This is a special case of a linking pairing.

Alexander Duality is the statement that if $X\subset S^n $ is compact and is a deformation retract of some open $U$ and $Y=S^n-X$ then $H_i(Y)$ is isomorphic to $H^{n-i-1}(X)$. The pairing giving this isomorphism is called the linking pairing, of which your pairing is a special case.

You are missing some regularity. You probably want $U$ to have a compact deformation retract. For instance the first homology of the complement of the Cantor set in the plane will have infinite rank.

The bit about compact support is because should be thinking of the set $U$ as a subset of the sphere, not a subset of the plane.

An obtuse but elementary development of this theorem can be found, at least for the plane, in the end of Dugundji's point set topology book. I learned Alexander duality from Spanier, (it might be dry, but its correct!). Generally, when I am teaching this to graduate students I illustrate it with practical computations, as the theorem itself is somewhat hard to get your head around.

You might like Kuga's "Galois' Dream" which covers a lot of connections between baby algebraic geometry and baby topology in the framework of Galois theory. Believe it or not, I really liked Lang's book on Algebraic curves as a place to learn about elementary consequences of duality in an algebraic setting. The Riemann-Roch theorem is the starting point for some of the most important results of the twentieth century, and it is a duality result much in the vein of what you seem to be interested in.