§ 12. Homology Groups of Chain Complexes

1968 ◽  
pp. 55-57
Keyword(s):  
Homology Groups ◽  
Chain Complexes

scholarly journals Khovanov Homology of Three-Strand Braid Links

Symmetry ◽  
2018 ◽  
Vol 10 (12) ◽  
pp. 720
Author(s):  
Young Kwun ◽  
Abdul Nizami ◽  
Mobeen Munir ◽  
Zaffar Iqbal ◽  
Dishya Arshad ◽  
...  
Keyword(s):  
Euler Characteristic ◽  
Jones Polynomial ◽  
Khovanov Homology ◽  
Homology Groups ◽  
Link Invariants ◽  
Chain Complexes ◽  
Chain Homotopy

Khovanov homology is a categorication of the Jones polynomial. It consists of graded chain complexes which, up to chain homotopy, are link invariants, and whose graded Euler characteristic is equal to the Jones polynomial of the link. In this article we give some Khovanov homology groups of 3-strand braid links Δ 2 k + 1 = x 1 2 k + 2 x 2 x 1 2 x 2 2 x 1 2 ⋯ x 2 2 x 1 2 x 1 2 , Δ 2 k + 1 x 2 , and Δ 2 k + 1 x 1 , where Δ is the Garside element x 1 x 2 x 1 , and which are three out of all six classes of the general braid x 1 x 2 x 1 x 2 ⋯ with n factors.

scholarly journals A spectral sequence for the homology of a finite algebraic delooping

2014 ◽  
Vol 13 (3) ◽  
pp. 563-599 ◽  
Author(s):  
Birgit Richter ◽  
Stephanie Ziegenhagen
Keyword(s):  
Topological Spaces ◽  
Loop Spaces ◽  
Polynomial Algebras ◽  
Symmetric Powers ◽  
Homology Groups ◽  
Chain Complexes ◽  
Kähler Differentials ◽  
The World ◽  
Derived Functors ◽  
Gerstenhaber Algebras

AbstractIn the world of chain complexes En-algebras are the analogues of based n-fold loop spaces in the category of topological spaces. Fresse showed that operadic En-homology of an En-algebra computes the homology of an n-fold algebraic delooping. The aim of this paper is to construct two spectral sequences for calculating these homology groups and to treat some concrete classes of examples such as Hochschild cochains, graded polynomial algebras and chains on iterated loop spaces. In characteristic zero we gain an identification of the summands in Pirashvili's Hodge decomposition of higher order Hochschild homology in terms of derived functors of indecomposables of Gerstenhaber algebras and as the homology of exterior and symmetric powers of derived Kähler differentials.

scholarly journals Homotopy classification of filtered complexes

1973 ◽  
Vol 15 (3) ◽  
pp. 298-318 ◽  
Author(s):  
Ross Street
Keyword(s):  
Abelian Groups ◽  
Classification Theorem ◽  
Homotopy Equivalence ◽  
Homotopy Classification ◽  
Homotopy Classes ◽  
Homology Groups ◽  
Chain Complexes ◽  
Kunneth Formula ◽  
Homology Functor

The homology functor from the category of free abelian chain complexes and homotopy classes of maps to that of graded abelian groups is full and replete (surjective on objects up to isomorphism) and reflects isomorphisms. Thus such a complex is determined to within homotopy equivalence (although not a unique homotopy equivalence) by its homology. The homotopy classes of maps between two such complexes should therefore be expressible in terms of the homology groups, and such an expression is in fact provided by the Künneth formula for Hom, sometimes called ‘the homotopy classification theorem’.

scholarly journals Profinite rigidity for twisted Alexander polynomials

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Jun Ueki
Keyword(s):  
Rigidity Theorem ◽  
Homology Groups ◽  
Alexander Polynomials ◽  
Twisted Homology ◽  
Hyperbolic Volumes

AbstractWe formulate and prove a profinite rigidity theorem for the twisted Alexander polynomials up to several types of finite ambiguity. We also establish torsion growth formulas of the twisted homology groups in a {{\mathbb{Z}}}-cover of a 3-manifold with use of Mahler measures. We examine several examples associated to Riley’s parabolic representations of two-bridge knot groups and give a remark on hyperbolic volumes.

scholarly journals Path homology theory of edge-colored graphs

2021 ◽  
Vol 19 (1) ◽  
pp. 706-723
Author(s):  
Yuri V. Muranov ◽  
Anna Szczepkowska
Keyword(s):  
Spectral Sequence ◽  
Edge Coloring ◽  
Homology Theory ◽  
Homotopy Category ◽  
Homology Groups ◽  
Colored Graphs ◽  
Natural Filtration ◽  
Definition Of ◽  
Exact Sequences

Abstract In this paper, we introduce the category and the homotopy category of edge-colored digraphs and construct the functorial homology theory on the foundation of the path homology theory provided by Grigoryan, Muranov, and Shing-Tung Yau. We give the construction of the path homology theory for edge-colored graphs that follows immediately from the consideration of natural functor from the category of graphs to the subcategory of symmetrical digraphs. We describe the natural filtration of path homology groups of any digraph equipped with edge coloring, provide the definition of the corresponding spectral sequence, and obtain commutative diagrams and braids of exact sequences.

Exponents for extraordinary homology groups

1993 ◽  
Vol 68 (1) ◽  
pp. 653-672 ◽  
Author(s):  
Dominique Arlettaz
Keyword(s):  
Homology Groups

Quasi-1D chains of dinickel lantern complexes and their magnetic properties

2017 ◽  
Vol 46 (17) ◽  
pp. 5546-5557 ◽  
Author(s):  
Arunpatcha Nimthong-Roldán ◽  
Jesse L. Guillet ◽  
James McNeely ◽  
Tarik J. Ozumerzifon ◽  
Matthew P. Shores ◽  
...  
Keyword(s):  
Magnetic Properties ◽  
Bridging Ligands ◽  
Chain Complexes ◽  
1D Chains

Four new quasi-1D Ni2lantern chain complexes of the form [Ni2(SOCR)4(L)]∞were prepared withN,N′-donor bridging ligands pyrazine and DABCO.

General position properties of ANR's

1982 ◽  
Vol 92 (3) ◽  
pp. 451-466 ◽  
Author(s):  
W. J. R. Mitchell
Keyword(s):  
Real Line ◽  
General Position ◽  
Cartesian Product ◽  
Vital Role ◽  
The Real ◽  
Homology Groups ◽  
Local Homology

This paper investigates the ‘general position’ properties which ANR's may possess. The most important of these is the disjoint discs property of Cannon (5), which plays a vital role in recent striking characterizations of manifolds (5, 9, 12, 18, 19, 22). Also considered are the property Δ of Borsuk(2) (which ensures an abundance of dimension-preserving maps), and the vanishing of local homology groups up to a given dimension (cf. (9)). Our main results give relations between these properties, and clarify their behaviour under the stabilization operation of taking cartesian product with the real line. In the last section these results are applied to give partial solutions to questions about homogeneous ANR's.

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