Heegaard Genus Formula for Haken Manifolds

Abstract

Suppose M is a compact orientable 3-manifold and Q ⊂ M a properly embedded orientable boundary incompressible essential surface. Denote the completions of the components of M -Q with respect to the path metric by M 1 , . . . , M k . Denote the smallest possible genus of a Heegaard splitting of M, or M j respectively, for which ∂ M, or ∂ M j respectively, is contained in one compression body by g(M, ∂ M), or g(M j , ∂ M j ) respectively. Denote the maximal number of non-parallel essential annuli that can be simultaneously embedded in M j by n j . ThenKeywords Heegaard genus · Essential surface · Genus formula Mathematics Subject Classification (2000) 57N10Heegaard splittings have long been used in the study of 3-manifolds. One reason for their continued importance in this study is that the Heegaard genus of a compact 3-manifold has proven to capture the topology of the 3-manifold more accurately than many other invariants. In particular, it provides an upper bound for the rank of the fundamental group of the 3-manifold, and this upper bound need not be sharp, as seen in the examples provided by Boileau and Zieschang [1]. We here prove the following: Let M be a compact orientable 3-manifold and Q ⊂ M an orientable boundary incompressible essential surface. Denote the completions of the components of M − Q with respect to the path metric by M 1 , . . . , M k . Denote the smallest possible genus of a Heegaard splitting of M, or M j respectively, for which ∂ M, or ∂ M j respectively, is contained in one compression body by g(M, ∂ M) or g(M j , ∂ M j ) respectively. Here g(M, ∂ M) is called the relative genus of M. Denote the maximal number of non parallel essential annuli that can be simultaneously embedded in M j by n j . Then

Heegaard splittings, the virtually Haken conjecture and Property (τ)

“…We obtain (⌊ i n ⌋ − 1) copies of M (n) and one copy of M (i − ⌊ i n ⌋n + n). Applying Theorem 4.4 of [53], we obtain…”

The Heegaard Genus of Amalgamated 3-Manifolds

“…Inequalities going in the other direction have been discovered by Johannson [3] and Schultens [12] who proved, respectively, that…”

Positivity of the universal pairing in 3 dimensions

“…By the assumption on the γ i , the manifolds DC n are all homology S 1 × S 2 's, so one cannot obtain a lower bound on their Heegaard genus from homological invariants. The point is not that this result cannot be obtained by other means (although for some cases of the above construction, the method of [41] also yields a lower bound which is optimal up to a constant); the point is that this estimate is obtained as a formal consequence of the properties of Dijkgraaf-Witten TQFTs. Classical 3-manifold topology enters only implicitly (but importantly) in the fact that π 1 (C) is residually finite.…”