The structure of a solvmanifold's Heegaard splittings

TL;DR

This paper classifies the isotopy classes of irreducible Heegaard splittings in solvmanifolds, revealing how the monodromy's trace influences the genus and reducibility of the splittings.

Contribution

It provides a complete classification of Heegaard splittings in solvmanifolds based on monodromy properties, including the number and type of irreducible splittings.

Findings

Irreducible splittings are strongly irreducible of genus two when trace is 3.

Splittings are isotopic if the trace is 4 or greater.

Splittings are weakly reducible of genus three when the monodromy cannot be expressed as specified.

Abstract

We classify isotopy classes of irreducible Heegaard splittings of solvmanifolds. If the monodromy of the solvmanifold can be expressed as a 2 x 2 matrix with 0 in the lower right hand corner (as always is true when the absolute value of the trace is 3), then any irreducible splitting is strongly irreducible and of genus two. If furthermore the absolute value of the trace is 4 or greater, then any two such splittings are isotopic. If the absolute value of the trace is 3 then, up to isotopy, there are exactly two irreducible splittings, their associated hyperelliptic involutions commute, and the product of the involutions is the central involution of the solvmanifold. If the monodromy cannot be expressed as a 2 x 2 matrix with 0 in the lower right hand corner, then the splitting is weakly reducible, of genus three and unique up to isotopy.