The complex of essential surfaces is contractible

Abstract

Let M be a 3-manifold and S be a component of ∂M. We introduce a simplicial complex E(M,S), whose vertices are isotopy classes of compact essential surfaces in M with at least one boundary component lying on S. E(M,S) can be thought as an analogy to the arc complex of a surface. We show that if M is irreducible and S is compressible in M, then E(M,S) is contractible.