On the quotients of mapping class groups of surfaces by the Johnson subgroups

2019 ◽  
pp. 1-23
Author(s):  
TOMÁŠ ZEMAN
Keyword(s):  
Boundary Component ◽  
Loop Space ◽  
Mapping Class Groups ◽  
Mapping Class ◽  
Classifying Spaces ◽  
Class Groups ◽  
Loop Spaces ◽  
Infinite Loop Spaces ◽  
Infinite Loop ◽  
Plus Construction

Abstract We study quotients of mapping class groups ${\Gamma _{g,1}}$ of oriented surfaces with one boundary component by the subgroups ${{\cal I}_{g,1}}(k)$ in the Johnson filtrations, and we show that the stable classifying spaces ${\mathbb {Z}} \times B{({\Gamma _\infty }/{{\cal I}_\infty }(k))^ + }$ after plus-construction are infinite loop spaces, fitting into a tower of infinite loop space maps that interpolates between the infinite loop spaces ${\mathbb {Z}} \times B\Gamma _\infty ^ + $ and ${\mathbb {Z}} \times B{({\Gamma _\infty }/{{\cal I}_\infty }(1))^ + } \simeq {\mathbb {Z}} \times B{\rm{Sp}}{({\mathbb {Z}})^ + }$ . We also show that for each level k of the Johnson filtration, the homology of these quotients with suitable systems of twisted coefficients stabilises as the genus of the surface goes to infinity.

A splitting for the stable mapping class group

1999 ◽  
Vol 127 (1) ◽  
pp. 55-65 ◽  
Author(s):  
ULRIKE TILLMANN
Keyword(s):  
Mapping Class Group ◽  
Class Group ◽  
Mapping Class ◽  
Loop Spaces ◽  
Classifying Space ◽  
Stable Homotopy Groups ◽  
Stable Mapping ◽  
Infinite Loop Spaces ◽  
Infinite Loop ◽  
Plus Construction

The main result of [15] is that the classifying space of the stable mapping class group after plus construction BΓ+∞ is an infinite loop space. This result is used to show that, localized away from two, a connected component of the stable homotopy groups of spheres QS0 splits off BΓ+∞. The splitting is a splitting of infinite loop spaces. It follows immediately that the homology with coefficients in ℤ[½] of the infinite symmetric group is a direct summand of the homology of the stable mapping class group.

scholarly journals ON HOMOLOGY OF VIRTUAL BRAIDS AND BURAU REPRESENTATION

2001 ◽  
Vol 10 (05) ◽  
pp. 795-812 ◽  
Author(s):  
VLADIMIR V. VERSHININ
Keyword(s):  
Infinite Number ◽  
Loop Space ◽  
Loop Spaces ◽  
Classifying Space ◽  
Burau Representation ◽  
Vassiliev Invariants ◽  
Virtual Knots ◽  
Infinite Loop Spaces ◽  
Infinite Loop ◽  
Plus Construction

Virtual knots arise in the study of Gauss diagrams and Vassiliev invariants of usual knots. The group of virtual braids on n strings VBn and its Burau representation to GLnℤ[t,t-1] also can be considered. The homological properties of the series of groups VBn and its Burau representation are studied. The following splitting of infinite loop spaces is proved for the plus-construction of the classifying space of the virtual braid group on the infinite number of strings: [Formula: see text] where Y is an infinite loop space. Connections with K*ℤ are discussed.

scholarly journals Homology stability for symmetric diffeomorphism and mapping class groups

2015 ◽  
Vol 160 (1) ◽  
pp. 121-139 ◽  
Author(s):  
ULRIKE TILLMANN
Keyword(s):  
Compact Manifold ◽  
Smooth Manifold ◽  
Mapping Class Groups ◽  
Mapping Class ◽  
Classifying Spaces ◽  
Class Groups ◽  
Connected Sum ◽  
Smooth Compact Manifold ◽  
Embedded Disks ◽  
Homology Stability

AbstractFor any smooth compact manifold W with boundary of dimension of at least two we prove that the classifying spaces of its group of diffeomorphisms which fix a set of k points or k embedded disks (up to permutation) satisfy homology stability. The same is true for so-called symmetric diffeomorphisms of W connected sum with k copies of an arbitrary compact smooth manifold Q of the same dimension. The analogues for mapping class groups as well as other generalisations will also be proved.

Infinite loop spaces with trivial Dyer-Lashof operations

1993 ◽  
Vol 113 (2) ◽  
pp. 311-328
Author(s):  
Michael Slack
Keyword(s):  
Loop Space ◽  
Homotopy Equivalent ◽  
Loop Spaces ◽  
Infinite Loop Spaces ◽  
Infinite Loop ◽  
Infinite Loop Space

AbstractLet p be any prime. It is well known that the modp Dyer-Lashof algebra acts trivially on the mod p homology of an Eilenberg-MacLane space. The main result of this paper is a converse of this fact. Specifically, it is shown that any connected infinite loop space with trivial action of the mod p Dyer-Lashof algebra is (localized at p) homotopy equivalent to a product of Eilenberg-MacLane spaces. It is then shown that this equivalence does not necessarily respect the infinite loop structures involved.

An Operad Action on Infinite Loop Space Multiplication

1977 ◽  
Vol 29 (6) ◽  
pp. 1208-1216 ◽  
Author(s):  
Thomas Lada
Keyword(s):  
Loop Space ◽  
Loop Spaces ◽  
Infinite Loop Spaces ◽  
Infinite Loop ◽  
Infinite Loop Space

It is well-known that an infinite loop space is an H-space whose multiplication enjoys nice properties concerning associativity and commutativity. A practical way of identifying infinite loop spaces is the utilization of May's recognition principle [3; 4]. To apply this principle, one requires an E∞-operad action on a space X; this action gives rise to various multiplications on X. In this note, it is shown that such multiplications enjoy an operad action up to homotopy that encodes the associativity and commutativity information, and that May's delooping theorem may be applied to them. We refer to [3] for the terminology of operads and monads.

scholarly journals THE PILLAR SWITCHINGS OF MAPPING CLASS GROUPS OF SURFACES

2013 ◽  
Vol 24 (13) ◽  
pp. 1350103 ◽  
Author(s):  
CHAN-SEOK JEONG ◽  
YONGJIN SONG
Keyword(s):  
Mapping Class Groups ◽  
Mapping Class ◽  
Double Loop ◽  
Stable Range ◽  
Class Groups ◽  
Loop Spaces ◽  
Dehn Twists

By gluing two copies of surface S0,g+2 along g + 1 holes, we get surface Sg,1. The pillar switching is a self-homeomorphism of Sg,1 which switches two pillars of surfaces by 180° horizontal rotation. We analyze the actions of the pillar switchings on π1Sg,1 and then give concrete expressions of the pillar switchings in terms of standard Dehn twists. The map ψ : Bg → Γg,1 sending the generators of Bg to the pillar switchings on Sg,1 is defined by extending the embedding Bg ↪ Γ0,(g+1),1. We show that this map is injective by analyzing the actions of pillar switchings on π1Sg,1. We also prove that this map induces a trivial homology homomorphism in the stable range. For the proof we use the categorical delooping method. We construct a suitable monoidal 2-functor from tile category to surface category and show that this functor thus induces a map of double loop spaces.

scholarly journals On the infinite loop spaces of algebraic cobordism and the motivic sphere

2021 ◽  
Vol Volume 5 ◽  
Author(s):  
Tom Bachmann ◽  
Elden Elmanto ◽  
Marc Hoyois ◽  
Adeel A. Khan ◽  
Vladimir Sosnilo ◽  
...  
Keyword(s):  
Finite Fields ◽  
Hilbert Scheme ◽  
Positive Characteristic ◽  
Loop Spaces ◽  
Final Version ◽  
Geometric Models ◽  
Infinite Loop Spaces ◽  
Infinite Loop ◽  
Algebraic Cobordism ◽  
Plus Construction

We obtain geometric models for the infinite loop spaces of the motivic spectra $\mathrm{MGL}$, $\mathrm{MSL}$, and $\mathbf{1}$ over a field. They are motivically equivalent to $\mathbb{Z}\times \mathrm{Hilb}_\infty^\mathrm{lci}(\mathbb{A}^\infty)^+$, $\mathbb{Z}\times \mathrm{Hilb}_\infty^\mathrm{or}(\mathbb{A}^\infty)^+$, and $\mathbb{Z}\times \mathrm{Hilb}_\infty^\mathrm{fr}(\mathbb{A}^\infty)^+$, respectively, where $\mathrm{Hilb}_d^\mathrm{lci}(\mathbb{A}^n)$ (resp. $\mathrm{Hilb}_d^\mathrm{or}(\mathbb{A}^n)$, $\mathrm{Hilb}_d^\mathrm{fr}(\mathbb{A}^n)$) is the Hilbert scheme of lci points (resp. oriented points, framed points) of degree $d$ in $\mathbb{A}^n$, and $+$ is Quillen's plus construction. Moreover, we show that the plus construction is redundant in positive characteristic. Comment: 13 pages. v5: published version; v4: final version, to appear in \'Epijournal G\'eom. Alg\'ebrique; v3: minor corrections; v2: added details in the moving lemma over finite fields

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