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On the geometric and algebraic rank of graph manifolds (2003)
| Citations: | 16 - 1 self |
BibTeX
@TECHREPORT{Schultens03onthe,
author = {Jennifer Schultens and Richard Weidmann},
title = {On the geometric and algebraic rank of graph manifolds},
institution = {},
year = {2003}
}
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Abstract
Abstract. For any n ∈ N we construct graph manifolds of genus 4n that have 3n-generated fundamental group. 1. introduction A Heegaard surface of an orientable closed 3-manifold M is an embedded orientable surface S such that M − S consists of 2 handlebodies V1 and V2. This decomposition of M is called a Heegaard splitting and denoted by M = V1 ∪S V2. We say that the splitting is of genus g if S is of genus g. It is not difficult to see that any orientable closed 3-manifold admits a Heegaard splitting. If M admits a Heegaard splitting of genus g but no Heegaard splitting of smaller genus then we say that M has Heegaard genus g and write g(M) = g. Clearly any curve in a handlebody can be homotoped to its boundary. It follows that for any Heegaard splitting M = V1 ∪S V2 every curve in M can be homotoped into V1. Thus the map induced by the inclusion of V1 into M maps a generating set of π1(V1) to a generating set of π1(M). As π1(V1) is generated by g elements it follows that π1(M) is also generated by g elements. Thus g(M) ≥ r(M) where
