A counterexample to the equivariant simple loop conjecture

Abstract

In 1985 Gabai gave a complete proof of the Simple Loop Conjecture, which states that any map between closed surfaces, which does not induce an injection on the level of π 1 {\pi _1} , takes some noncontractible simple loop in the domain surface to a contractible loop in the target surface. In this paper we study an analogous result for the category of surfaces equipped with finite group actions and the maps which commute with the group structures (the "equivariant" maps). We find a counterexample to the equivariant analog of the Simple Loop Conjecture for the cyclic group of order 3 3 . The proof uses an equivariant analog of a theorem of Edmonds which gives a standard geometric representative for any homotopy class of surface maps of nonzero degree.