The Magnus representation and higher-order Alexander invariants for homology cobordisms of surfaces

Abstract

803The Magnus representation and higher-order Alexander invariants for homology cobordisms of surfaces TAKUYA SAKASAIThe set of homology cobordisms from a surface to itself with markings of their boundaries has a natural monoid structure. To investigate the structure of this monoid, we define and study its Magnus representation and Reidemeister torsion invariants by generalizing Kirk, Livingston and Wang's argument over the Gassner representation of string links. Then, by applying Cochran and Harvey's framework of higher-order (noncommutative) Alexander invariants to them, we extract several information about the monoid and related objects.57M05; 57M27, 20F34, 57N05 IntroductionLet † g;1 be a compact connected oriented surface of genus g 1 with one boundary component. A homology cylinder over † g;1 consists of a homology cobordism from † g;1 to itself with markings of its boundary. We denote by C g;1 the set of isomorphisms classes of homology cylinders. Since stacking two homology cylinders gives a new one, we can endow C g;1 with a monoid structure (see Section 2 for the precise definition). The origin of homology cylinders goes back to Goussarov [10], Habiro [11], Garoufalidis-Levine [9] and Levine [21], where the clasper (or clover) surgery theory is effectively used to investigate the structure of C g;1 .As mentioned in [9] and [21], we can construct a homology cylinder from each homology 3-sphere or pure string link. Also, for a given homology cylinder, we can use an element of the mapping class group M g;1 of † g;1 to give another one by changing its markings. Since these operations preserve each monoid structure, C g;1 can be regarded as a simultaneous generalization of the monoid of homology 3-spheres, that of string links and M g;1 , any of which plays an important role in the theory of 3-manifolds.On the other hand, there exist some natural ways (see Section 6.3) to construct closed 3-manifolds from each homology cylinder. Therefore, by using its monoid structure, C g;1 serves as a tool for classifying closed 3-manifolds.The aim of this paper is to study the structure of C g;1 from an algebraic point of view. The main tools for that are invariants using noncommutative rings arising from group rings of fundamental groups of homology cylinders. The first half of this paper is occupied by defining the Magnus representation for homology cylinders and observing its fundamental properties. In the latter half, Cochran and Harvey's higherorder Alexander invariants are introduced and used to define numerical invariants for matrices obtained by the Magnus representation or associated Reidemeister torsions, whose entries are in noncommutative rings so that it is difficult to treat directly. This combination of the Magnus representation and higher-order Alexander invariants leads us to derive various facts about the structure of C g;1 and to give applications to the theory of closed 3-manifolds.The outline of this paper is as follows.In Section 2, we first recall the definition of homology cylinders over † g;1 by...

The Symplecticity of the Magnus Representation for Homology Cobordisms of Surfaces

“…Note that the statement of Theorem 2.4 has been already announced in [14], where we observe that the condition obtained from our theorem for a matrix to be written as the Magnus matrix of a homology cylinder is a strong one by considering a relationship to the theory of higher-order Alexander invariants.…”

“…Instead, we here explain that this result is obtained as a corollary of Theorem 2.4 when we restrict ra,k to VS£ n . By using a monoid homomorphism $ : VSC n -t C n ,i defined by Levine [7], we showed in [14] that holds for each pure string link L, where we identify 7 n + j e 7riE nil with <$, -6 ^\D n for i = 1 , . .…”

“…in [13,14], where Ks k is a certain skew field. This representation extends the Magnus representation for M g<x defined by Morita [8], as the Gassner representation for pure string links due to Le Dimet [6] and Kirk, Livingston and Wang [5] does that for the pure braid group.…”

Johnson-Morita theory in mapping class groups and monoids of homology cobordisms of surfaces

“…Their non-fiberedness was first shown by Friedl and Kim in [25] using twisted Alexander invariants. In [30], Goda and the author gave another proof of the nonfiberedness by using a Reidemeister torsion invariant discussed in [91]. (3) Pretzel knots P n = P(−2n + 1, 2n + 1, 2n 2 + 1) are homologically fibered knots of genus 1.…”

Equivalence relations for homology cylinders and the core of the Casson invariant

“…The Alexander polynomial and the Reidemeister-Turaev torsion. There is a relative version of the Alexander polynomial for homology cylinders [60]. The relative Alexander polynomial of an M ∈ IC is the order of the relative homology group…”