Heegaard genus formula for Haken manifolds

TL;DR

This paper establishes a new inequality relating the Heegaard genus of a 3-manifold with an incompressible surface to the genera of its decomposed components, incorporating topological invariants and essential annuli.

Contribution

It provides a novel formula connecting the Heegaard genus of Haken manifolds with their decompositions along incompressible surfaces.

Findings

Derived an inequality linking Heegaard genus of M and its components

Bound the sum of component genera using Euler characteristic and annuli classes

Enhanced understanding of manifold decomposition in 3-manifold topology

Abstract

Given a 3-manifold M containing an incompressible surface Q, we obtain an inequality relating the Heegaard genus of M and the Heegaard genera of the components of M - Q. Here the sum of the genera of the components of M - Q is bounded above by a linear expression in terms of the genus of M, the Euler characteristic of Q and the number of parallelism classes of essential annuli for which representatives can be simultaneously imbedded in the components of M - Q.