Equivariant Heegaard genus of reducible 3-manifolds

Abstract

The equivariant Heegaard genus of a 3-manifold M with the action of a finite group G of diffeomorphisms is the smallest genus of an equivariant Heegaard surface for M . Although a Heegaard splitting of a reducible manifold is reducible and although if M is reducible, there is an equivariant essential sphere, we show that equivariant Heegaard genus may be super-additive, additive, or sub-additive under equivariant connected sum. Using a thin position theory for 3dimensional orbifolds, we establish sharp bounds on the equivariant Heegaard genus of reducible manifolds, similar to those known for tunnel number.