On the fiber product over infinite-genus Riemann surfaces

TL;DR

This paper investigates the topological and connectedness properties of fiber products over infinite-genus Riemann surfaces, relating their ends space to the normal fiber product and analyzing conditions for connectedness.

Contribution

It establishes conditions linking the ends space of fiber products to the topological type of the involved Riemann surfaces and studies fiber products over infinite hyperelliptic curves.

Findings

Ends space of fiber product relates to that of the normal fiber product.

Conditions for connectedness of the fiber product are provided.

Analysis of fiber products over infinite hyperelliptic curves.

Abstract

Considering non-constant holomorphic maps $\beta_{i}:S_{i}\to S_{0}$, $i\in\{1,2\}$, between non-compact Riemann surfaces for which it is associated its fiber product $S_{1}\times_{(\beta_{1},\beta_{2})}S_{2}$. With this setting, in this paper we relate the ends space of such fiber product to the ends space of its normal fiber product. Moreover, we provide conditions on the maps $\beta_{1}$ and $\beta_{2}$ to guarantee connectednes on the fiber product. From these conditions, we link the ends space of fiber product with the topological type of the Riemann surfaces $S_{1}$ and $S_{2}$. We also study the fiber product over infinite hyperelliptic curves and discuss its connectedness and ends space.