scholarly journals The cobordism group of homology cylinders

2010 ◽  
Vol 147 (3) ◽  
pp. 914-942 ◽  
Author(s):  
Jae Choon Cha ◽  
Stefan Friedl ◽  
Taehee Kim
Keyword(s):  
Mapping Class Group ◽  
Betti Number ◽  
Boundary Component ◽  
Class Group ◽  
Mapping Class ◽  
Torelli Group ◽  
Cobordism Group ◽  
Infinite Rank ◽  
Homology Cobordism

AbstractGaroufalidis and Levine introduced the homology cobordism group of homology cylinders over a surface. This group can be regarded as an enlargement of the mapping class group. Using torsion invariants, we show that the abelianization of this group is infinitely generated provided that the first Betti number of the surface is positive. In particular, this shows that the group is not perfect. This answers questions of Garoufalidis and Levine, and Goda and Sakasai. Furthermore, we show that the abelianization of the group has infinite rank for the case that the surface has more than one boundary component. These results also hold for the homology cylinder analogue of the Torelli group.

scholarly journals Johnson–Levine homomorphisms and the tree reduction of the LMO functor

2019 ◽  
pp. 1-35
Author(s):  
ANDERSON VERA
Keyword(s):  
Mapping Class Group ◽  
Boundary Component ◽  
Class Group ◽  
Mapping Class ◽  
Torelli Group ◽  
Oriented Surface ◽  
Topological Interpretation ◽  
Natural Way

Abstract Let $\mathcal{M}$ denote the mapping class group of Σ, a compact connected oriented surface with one boundary component. The action of $\mathcal{M}$ on the nilpotent quotients of π1(Σ) allows to define the so-called Johnson filtration and the Johnson homomorphisms. J. Levine introduced a new filtration of $\mathcal{M}$ , called the Lagrangian filtration. He also introduced a version of the Johnson homomorphisms for this new filtration. The first term of the Lagrangian filtration is the Lagrangian mapping class group, whose definition involves a handlebody bounded by Σ, and which contains the Torelli group. These constructions extend in a natural way to the monoid of homology cobordisms. Besides, D. Cheptea, K. Habiro and G. Massuyeau constructed a functorial extension of the LMO invariant, called the LMO functor, which takes values in a category of diagrams. In this paper we give a topological interpretation of the upper part of the tree reduction of the LMO functor in terms of the homomorphisms defined by J. Levine for the Lagrangian mapping class group. We also compare the Johnson filtration with the filtration introduced by J. Levine.

scholarly journals A lower bound of the distortion of the Torelli group in the mapping class group with boundary components

2017 ◽  
Vol 26 (08) ◽  
pp. 1750049
Author(s):  
Erika Kuno ◽  
Genki Omori
Keyword(s):  
Lower Bound ◽  
Mapping Class Group ◽  
Class Group ◽  
Mapping Class ◽  
Torelli Group ◽  
Oriented Surface

We prove that the Torelli group of an oriented surface with any number of boundary components is at least exponentially distorted in the mapping class group by using Broaddus–Farb–Putman’s techniques. Further we show that the distortion of the Torelli group in the level [Formula: see text] mapping class group is the same as that in the mapping class group.

scholarly journals The mapping class group action on -character varieties

2020 ◽  
pp. 1-15
Author(s):  
WILLIAM M. GOLDMAN ◽  
SEAN LAWTON ◽  
EUGENE Z. XIA
Keyword(s):  
Group Action ◽  
Mapping Class Group ◽  
Boundary Component ◽  
Class Group ◽  
Orientable Surface ◽  
Mapping Class ◽  
Character Variety ◽  
Character Varieties ◽  
Compact Orientable Surface

Let $\unicode[STIX]{x1D6F4}$ be a compact orientable surface of genus $g=1$ with $n=1$ boundary component. The mapping class group $\unicode[STIX]{x1D6E4}$ of $\unicode[STIX]{x1D6F4}$ acts on the $\mathsf{SU}(3)$ -character variety of $\unicode[STIX]{x1D6F4}$ . We show that the action is ergodic with respect to the natural symplectic measure on the character variety.

scholarly journals ABELIAN COVERS OF SURFACES AND THE HOMOLOGY OF THE LEVEL L MAPPING CLASS GROUP

2011 ◽  
Vol 03 (03) ◽  
pp. 265-306 ◽  
Author(s):  
ANDREW PUTMAN
Keyword(s):  
Group Ring ◽  
Mapping Class Group ◽  
Boundary Component ◽  
Homology Group ◽  
Class Group ◽  
Marked Point ◽  
Mapping Class ◽  
Rational Group ◽  
Rational Homology ◽  
Abelian Covers

We calculate the first homology group of the mapping class group with coefficients in the first rational homology group of the universal abelian ℤ/L-cover of the surface. If the surface has one marked point, then the answer is ℚτ(L), where τ(L) is the number of positive divisors of L. If the surface instead has one boundary component, then the answer is ℚ. We also perform the same calculation for the level L subgroup of the mapping class group. Set HL = H1(Σg; ℤ/L). If the surface has one marked point, then the answer is ℚ[HL], the rational group ring of HL. If the surface instead has one boundary component, then the answer is ℚ.

scholarly journals The Johnson homomorphism and its kernel

2018 ◽  
Vol 2018 (735) ◽  
pp. 109-141 ◽  
Author(s):  
Andrew Putman
Keyword(s):  
Finite Index ◽  
General Result ◽  
Mapping Class Group ◽  
Class Group ◽  
Mapping Class ◽  
Torelli Group ◽  
Finite Index Subgroups

AbstractWe give a new proof of a celebrated theorem of Dennis Johnson that asserts that the kernel of the Johnson homomorphism on the Torelli subgroup of the mapping class group is generated by separating twists. In fact, we prove a more general result that also applies to “subsurface Torelli groups”. Using this, we extend Johnson’s calculation of the rational abelianization of the Torelli group not only to the subsurface Torelli groups, but also to finite-index subgroups of the Torelli group that contain the kernel of the Johnson homomorphism.

scholarly journals Minimal Asymptotic Translation Lengths of Torelli Groups and Pure Braid Groups on the Curve Graph

2018 ◽  
Vol 2020 (24) ◽  
pp. 9974-9987
Author(s):  
Hyungryul Baik ◽  
Hyunshik Shin
Keyword(s):  
Braid Group ◽  
Mapping Class Group ◽  
Class Group ◽  
Braid Groups ◽  
Mapping Class ◽  
Torelli Group ◽  
Pure Braid Group ◽  
Curve Graph ◽  
Translation Length

Abstract In this paper, we show that the minimal asymptotic translation length of the Torelli group ${\mathcal{I}}_g$ of the surface $S_g$ of genus $g$ on the curve graph asymptotically behaves like $1/g$, contrary to the mapping class group ${\textrm{Mod}}(S_g)$, which behaves like $1/g^2$. We also show that the minimal asymptotic translation length of the pure braid group ${\textrm{PB}}_n$ on the curve graph asymptotically behaves like $1/n$, contrary to the braid group ${\textrm{B}}_n$, which behaves like $1/n^2$.

scholarly journals SIMPLY INTERSECTING PAIR MAPS IN THE MAPPING CLASS GROUP

2012 ◽  
Vol 21 (11) ◽  
pp. 1250107
Author(s):  
LEAH R. CHILDERS
Keyword(s):  
Mapping Class Group ◽  
Class Group ◽  
Mapping Class ◽  
Infinite Index ◽  
Index Subgroup ◽  
Torelli Group ◽  
Topological Characterization

The Torelli group, [Formula: see text], is the subgroup of the mapping class group consisting of elements that act trivially on the homology of the surface. There are three types of elements that naturally arise in studying [Formula: see text]: bounding pair maps, separating twists, and simply intersecting pair maps (SIP-maps). Historically the first two types of elements have been the focus of the literature on [Formula: see text], while SIP-maps have received relatively little attention until recently, due to an infinite presentation of [Formula: see text] introduced by Putman that uses all three types of elements. We will give a topological characterization of the image of an SIP-map under the Johnson homomorphism and Birman–Craggs–Johnson homomorphism. We will also classify which SIP-maps are in the kernel of these homomorphisms. Then we will look at the subgroup generated by all SIP-maps, SIP (Sg), and show it is an infinite index subgroup of [Formula: see text].

THE RESHETIKHIN-TURAEV REPRESENTATION OF THE MAPPING CLASS GROUP

1994 ◽  
Vol 03 (04) ◽  
pp. 547-574 ◽  
Author(s):  
GRETCHEN WRIGHT
Keyword(s):  
Mapping Class Group ◽  
Quadratic Forms ◽  
Class Group ◽  
Orientable Surface ◽  
Mapping Class ◽  
Representation Space ◽  
Torelli Group ◽  
Spin Structures ◽  
Basis Vectors

The Reshetikhin-Turaev representation of the mapping class group of an orientable surface is computed explicitly in the case r = 4. It is then shown that the restriction of this representation to the Torelli group is equal to the sum of the Birman-Craggs homomorphisms. The proof makes use of an explicit correspondence between the basis vectors of the representation space, and the Z/2Z-quadratic forms on the first homology of the surface. This result corresponds to the fact, shown by Kirby and Melvin, that the three-manifold invariant when r = 4 is related to spin structures on the associated four-manifold.

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