Taut sutured handlebodies as twisted homology products
Margaret Nichols (Chicago)
A basic problem in the study of 3-manifolds is to determine when geometric objects are of ‘minimal complexity’. We are interested in this question in the setting of sutured manifolds, where minimal complexity is called ‘tautness’.
One method for certifying that a sutured manifold is taut is to show that it is homologically simple - a so-called ‘rational homology product’. Most sutured manifolds do not have this form, but do always take the more general form of a ‘twisted homology product’, which incorporates a representation of the fundamental group. The question then becomes, how complicated of a representation is needed to realize a given sutured manifold as such.
We explore some classes of relatively simple sutured manifolds, and see even in the simplest of these, twisting is required. We give examples that, when restricted to representations of a particular stripe, the twisting representation cannot be ‘too simple’. We also discuss why we expect two-dimensional representations to suffice for these classes.