The Parametric Oka Principle for Riemann Surfaces

Abstract

In 1993, Winkelmann classified the pairs of Riemann surfaces which satisfy the basic Oka principle (BOP). We generalise Winkelmann’s result to include the notion of the parametric Oka principle (POP). Using low-dimensional techniques from algebraic topology and Riemann surface theory, we provide accessible proofs of POP for all pairs of Riemann surfaces satisfying BOP, besides the case of an open Riemann surface mapping into the Riemann sphere. For this case, we provide partial results. Winkelmann also provided a list of the pairs of Riemann surfaces which fail to satisfy BOP. To explore these pairs, we introduce the notion of the higher parametric Oka principle (hPOP). This is our own definition and is one of the main original contribution of this thesis. For Winkelmann’s counterexamples (labelled (i)–(v)), we ask whether they satisfy hPOP. We provide a counterexample for case (i), showing hPOP fails. For cases (ii), (iv) and (v), we provide full proofs showing hPOP holds. For case (iii), we provide partial affirmative results of hPOP.