Abstract

We use elementary algebraic methods to reprove a theorem which was proved by Pop using rigid analytic geometry and in a less general form by Harbater using formal algebraic patching:

Let C be an algebraically closed field of cardinality m. Consider a subset S of P¹(C) of cardinality m. Then the fundamental group of P¹(C)\ S is isomorphic to the free profinite group of rank m.

We also observe that if char(C)≠0 and 0 < card(S) < m, then π₁(P¹(C)\ S) is not isomorphic to a free profinite group.

Authors