In Ahlfor's "Complex Analaysis" it is stated (page 147) that:
"Every region (i.e. an open connected subset of $\Bbb C$) with a finite homology basis has finite connectivity and the number of basis elements is one less than the connectivity"
I have two problems with this statement:
- I am not sure what is the meaning to assign to homology basis in this context for Ahlfors only defined it on a region that was assumed to have finite connectivity; my best guess, especially in light of the discussione that followed that definition, is that by finite homology basis of a region $\Omega$ here is meant a finite family $\{\gamma_j\}_{j=1,...,n}$ of cycles such that any other cycle $\gamma \subseteq \Omega$ is homologous modulo $\Omega$ to a unique linear combinations of the basis elements with integer coefficients (I am not sure if it is necessary to add that the index of a basis element with respect to any point outside of $\Omega$ is either $0$ or $1$ because this is the property required for the construction of the basis when we already know that the region has finite connectivity);
- I have no idea on how to prove it (I tried something by contradiction but it becomes very difficult very fast because it seems to me that you need to do some long construction as in the previous theorems despite the statement being thrown in the book as if it were trivial).
I'd like to hear if anyone else also thought of this statement and if he/she had better judgement than me.