Obstructions to Lie–Rinehart Algebra Extensions

2011 ◽  
Vol 18 (01) ◽  
pp. 83-104 ◽  
Author(s):  
J. M. Casas
Keyword(s):  
Exact Sequence ◽  
Obstruction Theory

The problem of the representation of an action of a Lie–Rinehart algebra on a Lie 𝖠-algebra by means of a homomorphism of Lie–Rinehart algebras is studied. An eight-term exact sequence associated to an epimorphism of Lie–Rinehart algebras for the cohomology of Lie–Rinehart algebras developed by Casas, Ladra and Pirashvili is obtained. This sequence is applied to study the obstruction theory of Lie–Rinehart algebra extensions.

scholarly journals Comparing homotopy categories

2007 ◽  
Vol 2 (1) ◽  
pp. 169-205 ◽  
Author(s):  
David Blanc
Keyword(s):  
Exact Sequence ◽  
Obstruction Theory ◽  
Homotopy Groups ◽  
Model Categories

AbstractGiven a suitable functor T : → between model categories, we define a long exact sequence relating the homotopy groups of any X ε with those of TX, and use this to describe an obstruction theory for lifting an object G ε to . Examples include finding spaces with given homology or homotopy 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.

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.

scholarly journals The integral cohomology of the Hilbert scheme of points on a surface

2020 ◽  
Vol 8 ◽  
Author(s):  
Burt Totaro
Keyword(s):  
Natural Number ◽  
Hilbert Scheme ◽  
Obstruction Theory ◽  
Projective Surface ◽  
Hilbert Schemes ◽  
Complex Projective Surface ◽  
Smooth Complex ◽  
Hilbert Scheme Of Points ◽  
Complex Projective ◽  
Torsion Free

Abstract We show that if X is a smooth complex projective surface with torsion-free cohomology, then the Hilbert scheme $X^{[n]}$ has torsion-free cohomology for every natural number n. This extends earlier work by Markman on the case of Poisson surfaces. The proof uses Gholampour-Thomas’s reduced obstruction theory for nested Hilbert schemes of surfaces.

scholarly journals Properties of the Secondary Hochschild Homology

2018 ◽  
Vol 25 (02) ◽  
pp. 225-242
Author(s):  
Jacob Laubacher
Keyword(s):  
Exact Sequence ◽  
Morita Equivalence ◽  
Hochschild Homology ◽  
Low Dimension

In this paper we study properties of the secondary Hochschild homology of the triple (A, B, ε) with coefficients in M. We establish a type of Morita equivalence between two triples and show that H•((A, B, ε); M) is invariant under this equivalence. We also prove the existence of an exact sequence which connects the usual and the secondary Hochschild homologies in low dimension, allowing one to perform easy computations. The functoriality of H•((A, B, ε); M) is also discussed.

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