scholarly journals THE SECOND HOMOLOGY GROUP OF THE LEVEL 2 MAPPING CLASS GROUP AND EXTENDED TORELLI GROUP OF AN ORIENTABLE SURFACE

Topology ◽  
1999 ◽  
Vol 38 (6) ◽  
pp. 1175-1207 ◽  
Author(s):  
Joel Foisy
Keyword(s):  
Mapping Class Group ◽  
Homology Group ◽  
Class Group ◽  
Orientable Surface ◽  
Mapping Class ◽  
Torelli Group ◽  
Level 2

scholarly journals A minimal generating set of the level 2 mapping class group of a non-orientable surface

2014 ◽  
Vol 157 (2) ◽  
pp. 345-355
Author(s):  
SUSUMU HIROSE ◽  
MASATOSHI SATO
Keyword(s):  
Mapping Class Group ◽  
Class Group ◽  
Orientable Surface ◽  
Mapping Class ◽  
Nonorientable Surface ◽  
Generating Set ◽  
Level 2 ◽  
Minimal Generating Set

AbstractWe construct a minimal generating set of the level 2 mapping class group of a nonorientable surface of genus g, and determine its abelianization for g ≥ 4.

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.

First homology group of mapping class groups of nonorientable surfaces

1998 ◽  
Vol 123 (3) ◽  
pp. 487-499 ◽  
Author(s):  
MUSTAFA KORKMAZ
Keyword(s):  
Mapping Class Group ◽  
Homology Group ◽  
Class Group ◽  
Mapping Class Groups ◽  
Orientable Surface ◽  
Mapping Class ◽  
Class Groups ◽  
Homology Groups ◽  
Nonorientable Surfaces ◽  
Orientable Surfaces

Recall that the first homology group H1(G) of a group G is the derived quotient G/[G, G]. The first homology groups of the mapping class groups of closed orientable surfaces are well known. Let F be a closed orientable surface of genus g. Recall that the extended mapping class group [Mscr ]*F of the surface F is the group of the isotopy classes of self-homeomorphisms of F. The mapping class group [Mscr ]F of F is the subgroup of [Mscr ]*F consisting of the isotopy classes of orientation-preserving self-homeomorphisms of F. It is well known that [Mscr ]F is trivial if F is a sphere. Hence the first homology group of the mapping class group of a sphere is trivial. If the genus of F is at least three, then H1([Mscr ]F) is again trivial. This result is due to Powell [P]. The group H1([Mscr ]F) is Z10 if the genus of F is two, proved by Mumford [Mu], and Z12 if F is a torus. When a problem about orientable surfaces is solved, it is natural to ask the corresponding problem for nonorientable surfaces. This is our motivation for the present paper.

scholarly journals A combinatorial formula for Earle's twisted 1-cocycle on the mapping class group

2009 ◽  
Vol 146 (1) ◽  
pp. 109-118 ◽  
Author(s):  
YUSUKE KUNO
Keyword(s):  
Mapping Class Group ◽  
Homology Group ◽  
Class Group ◽  
Base Point ◽  
Mapping Class ◽  
Combinatorial Formula ◽  
Hyperelliptic Involution ◽  
Fixed Base ◽  
Genus 2 ◽  
Closed Oriented Surface

AbstractWe present a formula expressing Earle's twisted 1-cocycle on the mapping class group of a closed oriented surface of genus ≥ 2 relative to a fixed base point, with coefficients in the first homology group of the surface. For this purpose we compare it with Morita's twisted 1-cocycle which is combinatorial. The key is the computation of these cocycles on a particular element of the mapping class group, which is topologically a hyperelliptic involution.

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 On the Cohomology of the Mapping Class Group of the Punctured Projective Plane

2020 ◽  
Vol 71 (2) ◽  
pp. 539-555
Author(s):  
Miguel A Maldonado ◽  
Miguel A Xicoténcatl
Keyword(s):  
Exact Sequence ◽  
Configuration Space ◽  
Mapping Class Group ◽  
Homotopy Theory ◽  
Class Group ◽  
Orientable Surface ◽  
Configuration Spaces ◽  
Mapping Class ◽  
Serre Spectral Sequence ◽  
Classical Homotopy

Abstract The mapping class group $\Gamma ^k(N_g)$ of a non-orientable surface with punctures is studied via classical homotopy theory of configuration spaces. In particular, we obtain a non-orientable version of the Birman exact sequence. In the case of ${\mathbb{R}} \textrm{P}^2$, we analyze the Serre spectral sequence of a fiber bundle $F_k({\mathbb{R}}{\textrm{P}}^{2}) / \Sigma _k \to X_k \to BSO(3)$ where $X_k$ is a $K(\Gamma ^k({\mathbb{R}} \textrm{P}^2),1)$ and $F_k({\mathbb{R}}{\textrm{P}}^{2}) / \Sigma _k$ denotes the configuration space of unordered $k$-tuples of distinct points in ${\mathbb{R}} \textrm{P}^2$. As a consequence, we express the mod-2 cohomology of $\Gamma ^k({\mathbb{R}} \textrm{P}^2)$ in terms of that of $F_k({\mathbb{R}}{\textrm{P}}^{2}) / \Sigma _k$.

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.

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