scholarly journals ABELIAN CYCLES IN THE HOMOLOGY OF THE TORELLI GROUP

2021 ◽  
pp. 1-24
Author(s):  
Erik Lindell
Keyword(s):  
Boundary Component ◽  
Analogous Result ◽  
Marked Point ◽  
Stable Range ◽  
Torelli Group ◽  
Rational Homology ◽  
Oriented Surface ◽  
Homology Groups

Abstract In the early 1980s, Johnson defined a homomorphism $\mathcal {I}_{g}^1\to \bigwedge ^3 H_1\left (S_{g},\mathbb {Z}\right )$ , where $\mathcal {I}_{g}^1$ is the Torelli group of a closed, connected, and oriented surface of genus g with a boundary component and $S_g$ is the corresponding surface without a boundary component. This is known as the Johnson homomorphism. We study the map induced by the Johnson homomorphism on rational homology groups and apply it to abelian cycles determined by disjoint bounding-pair maps, in order to compute a large quotient of $H_n\left (\mathcal {I}_{g}^1,\mathbb {Q}\right )$ in the stable range. This also implies an analogous result for the stable rational homology of the Torelli group $\mathcal {I}_{g,1}$ of a surface with a marked point instead of a boundary component. Further, we investigate how much of the image of this map is generated by images of such cycles and use this to prove that in the pointed case, they generate a proper subrepresentation of $H_n\left (\mathcal {I}_{g,1}\right )$ for $n\ge 2$ and g large enough.

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 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.

IWASAWA TYPE FORMULA FOR COVERS OF A LINK IN A RATIONAL HOMOLOGY SPHERE

2008 ◽  
Vol 17 (10) ◽  
pp. 1199-1221 ◽  
Author(s):  
TERUHISA KADOKAMI ◽  
YASUSHI MIZUSAWA
Keyword(s):  
Number Field ◽  
Class Number ◽  
Homology Sphere ◽  
Rational Homology ◽  
Homology Groups ◽  
Cyclic Covers ◽  
Class Number Formula ◽  
Number Formula ◽  
Iwasawa Invariants ◽  
Rational Homology Sphere

Based on the analogy between links and primes, we present an analogue of the Iwasawa's class number formula in a Zp-extension for the p-homology groups of pn-fold cyclic covers of a link in a rational homology 3-sphere. We also describe the associated Iwasawa invariants precisely for some examples and discuss analogies with the number field case.

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 Homological Stability for Spaces of Commuting Elements in Lie Groups

2020 ◽  
Author(s):  
Daniel A Ramras ◽  
Mentor Stafa
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Stable Range ◽  
Rational Homology ◽  
Infinite Dimensional ◽  
Fixed Group ◽  
Character Varieties ◽  
Representation Stability ◽  
Equivariant Homology ◽  
Homological Stability

Abstract In this paper, we study homological stability for spaces $\textrm{Hom}({{\mathbb{Z}}}^n,G)$ of pairwise commuting $n$-tuples in a Lie group $G$. We prove that for each $n\geqslant 1$, these spaces satisfy rational homological stability as $G$ ranges through any of the classical sequences of compact, connected Lie groups, or their complexifications. We prove similar results for rational equivariant homology, for character varieties, and for the infinite-dimensional analogues of these spaces, $\textrm{Comm}(G)$ and $B_{\textrm{com}} G$, introduced by Cohen–Stafa and Adem–Cohen–Torres-Giese, respectively. In addition, we show that the rational homology of the space of unordered commuting $n$-tuples in a fixed group $G$ stabilizes as $n$ increases. Our proofs use the theory of representation stability—in particular, the theory of $\textrm{FI}_W$-modules developed by Church–Ellenberg–Farb and Wilson. In all of the these results, we obtain specific bounds on the stable range, and we show that the homology isomorphisms are induced by maps of spaces.

scholarly journals Mapping class group representations from Drinfeld doubles of finite groups

2020 ◽  
Vol 29 (05) ◽  
pp. 2050033
Author(s):  
Jens Fjelstad ◽  
Jürgen Fuchs
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Class Group ◽  
Marked Point ◽  
Mapping Class ◽  
Group Representations ◽  
Torelli Group ◽  
Group Data ◽  
A Finite Group ◽  
Marked Points

We investigate representations of mapping class groups of surfaces that arise from the untwisted Drinfeld double of a finite group [Formula: see text], focusing on surfaces without marked points or with one marked point. We obtain concrete descriptions of such representations in terms of finite group data. This allows us to establish various properties of these representations. In particular, we show that they have finite images, and that for surfaces of genus at least [Formula: see text] their restriction to the Torelli group is non-trivial if and only if [Formula: see text] is non-abelian.

Pseudo-immersions of oriented surfaces with one boundary component into the plane

2009 ◽  
Vol 139 (6) ◽  
pp. 1327-1335
Author(s):  
Minoru Yamamoto
Keyword(s):  
Boundary Component ◽  
Boundary Curve ◽  
Winding Number ◽  
Singular Set ◽  
Oriented Surface ◽  
Smooth Map

Let M be a compact connected oriented surface with exactly one boundary component. A pseudo-immersion of M into the plane is a smooth map such that it has only fold singularities and is an orientation-preserving immersion near the boundary of M. For a pseudo-immersion we study the relation between the winding number of the boundary curve and the number of the singular set components.

scholarly journals The Torelli group and the Kauffman bracket skein module

2017 ◽  
Vol 165 (1) ◽  
pp. 163-178
Author(s):  
SHUNSUKE TSUJI
Keyword(s):  
Torelli Group ◽  
Oriented Surface ◽  
Skein Module ◽  
Kauffman Bracket ◽  
Kauffman Bracket Skein Module ◽  
New Construction

AbstractWe introduce an embedding of the Torelli group of a compact connected oriented surface with non-empty connected boundary into the completed Kauffman bracket skein algebra of the surface, which gives a new construction of the first Johnson homomorphism.

scholarly journals Characterization of Y2-Equivalence for Homology Cylinders

2003 ◽  
Vol 12 (04) ◽  
pp. 493-522 ◽  
Author(s):  
Gwénaël Massuyeau ◽  
Jean-Baptiste Meilhan
Keyword(s):  
Abelian Group ◽  
Equivalence Relation ◽  
Boundary Component ◽  
Oriented Surface

For Σ a compact connected oriented surface, we consider homology cylinders over Σ: these are homology cobordisms with an extra homological triviality condition. When considered up to Y2-equivalence, which is a surgery equivalence relation arising from the Goussarov-Habiro theory, homology cylinders form an Abelian group. In this paper, when Σ has one or zero boundary component, we define a surgery map from a certain space of graphs to this group. This map is shown to be an isomorphism, with inverse given by some extensions of the first Johnson homomorphism and Birman-Craggs homomorphisms.

scholarly journals Rational homology and homotopy of high-dimensional string links

2018 ◽  
Vol 30 (5) ◽  
pp. 1209-1235
Author(s):  
Paul Arnaud Songhafouo Tsopméné ◽  
Victor Turchin
Keyword(s):  
Finite Graph ◽  
High Dimensional ◽  
Homotopy Groups ◽  
Maximal Dimension ◽  
Stable Range ◽  
Rational Homotopy ◽  
Rational Homology ◽  
Graph Complexes ◽  
Higher Dimensional ◽  
String Links

AbstractArone and the second author showed that when the dimensions are in the stable range, the rational homology and homotopy of the high-dimensional analogues of spaces of long knots can be calculated as the homology of a direct sum of finite graph-complexes that they described explicitly. They also showed that these homology and homotopy groups can be interpreted as the higher-order Hochschild homology, also called Hochschild–Pirashvili homology. In this paper, we generalize all these results to high-dimensional analogues of spaces of string links. The methods of our paper are applicable in the range when the ambient dimension is at least twice the maximal dimension of a link component plus two, which in particular guarantees that the spaces under study are connected. However, we conjecture that our homotopy graph-complex computes the rational homotopy groups of link spaces always when the codimension is greater than two, i.e. always when the Goodwillie–Weiss calculus is applicable. Using Haefliger’s approach to calculate the groups of isotopy classes of higher-dimensional links, we confirm our conjecture at the level of {\pi_{0}}.

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