scholarly journals An 8-periodic exact sequence of Witt groups of Azumaya algebras with involution

2022 ◽  
Author(s):  
Uriya A. First
Keyword(s):  
Exact Sequence ◽  
Azumaya Algebras ◽  
Witt Groups

Generating the Mapping Class Group

2017 ◽  
Author(s):  
Benson Farb ◽  
Dan Margalit
Keyword(s):  
Exact Sequence ◽  
Simplicial Complex ◽  
Mapping Class Group ◽  
Class Group ◽  
Mapping Class ◽  
Detailed Structure ◽  
Inductive Step ◽  
Closed Curves ◽  
Combinatorial Object ◽  
Generating Sets

This chapter considers the Dehn–Lickorish theorem, which states that when g is greater than or equal to 0, the mapping class group Mod(Sɡ) is generated by finitely many Dehn twists about nonseparating simple closed curves. The theorem is proved by induction on genus, and the Birman exact sequence is introduced as the key step for the induction. The key to the inductive step is to prove that the complex of curves C(Sɡ) is connected when g is greater than or equal to 2. The simplicial complex C(Sɡ) is a useful combinatorial object that encodes intersection patterns of simple closed curves in Sɡ. More detailed structure of C(Sɡ) is then used to find various explicit generating sets for Mod(Sɡ), including those due to Lickorish and to Humphries.

scholarly journals The resolution property via Azumaya algebras

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Siddharth Mathur
Keyword(s):  
Azumaya Algebras ◽  
Algebraic Space ◽  
Local Methods ◽  
Artin Stacks ◽  
Resolution Property

Abstract Using formal-local methods, we prove that a separated and normal tame Artin surface has the resolution property. By proving that normal tame Artin stacks can be rigidified, we ultimately reduce our analysis to establishing the existence of Azumaya algebras. Our construction passes through the case of tame Artin gerbes, tame Artin curves, and algebraic space surfaces, each of which we establish independently.

WHITEHEAD EXACT SEQUENCE AND DIFFERENTIAL GRADED FREE LIE ALGEBRA

2004 ◽  
Vol 15 (10) ◽  
pp. 987-1005 ◽  
Author(s):  
MAHMOUD BENKHALIFA
Keyword(s):  
Exact Sequence ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Integral Domain ◽  
Free Lie Algebra ◽  
Universal Algebras ◽  
Free Lie Algebras ◽  
Equivalent Class

Let R be a principal and integral domain. We say that two differential graded free Lie algebras over R (free dgl for short) are weakly equivalent if and only if the homologies of their corresponding enveloping universal algebras are isomophic. This paper is devoted to the problem of how we can characterize the weakly equivalent class of a free dgl. Our tool to address this question is the Whitehead exact sequence. We show, under a certain condition, that two R-free dgls are weakly equivalent if and only if their Whitehead sequences are isomorphic.

scholarly journals A Birman exact sequence for the Torelli subgroup of Aut(Fn)

2016 ◽  
Vol 26 (03) ◽  
pp. 585-617 ◽  
Author(s):  
Matthew Day ◽  
Andrew Putman
Keyword(s):  
Exact Sequence ◽  
Recent Work ◽  
Homology Group ◽  
Entire Group ◽  
Torelli Group

We develop an analogue of the Birman exact sequence for the Torelli subgroup of [Formula: see text]. This builds on earlier work of the authors, who studied an analogue of the Birman exact sequence for the entire group [Formula: see text]. These results play an important role in the authors’ recent work on the second homology group of the Torelli group.

Sign in / Sign up

Export Citation Format

Share Document