Single-space axioms for homology theory

1959 ◽  
Vol 55 (1) ◽  
pp. 10-22 ◽  
Author(s):  
G. M. Kelly
Keyword(s):  
Homology Theory ◽  
Homology Groups ◽  
Relative Homology ◽  
The Absolute

Eilenberg and Steenrod(1) give a set of axioms for the homology theory of pairs of spaces and their maps, and prove that these axioms are categorical on triangulable pairs.Here we give a set of axioms for the homology theory of single spaces and their maps, that is, for absolute rather than relative homology. This axiomatization is shown to be essentially equivalent to that of Eilenberg and Steenrod, the relative homology groups being suitably denned in terms of the absolute groups.

Singular homology theory in the category of soft topological spaces

2015 ◽  
Vol 22 (4) ◽  
Author(s):  
Sadi Bayramov ◽  
Cigdem Gündüz (Aras) ◽  
Leonard Mdzinarishvili
Keyword(s):  
Homology Group ◽  
Homology Theory ◽  
Topological Spaces ◽  
Homology Groups ◽  
Relative Homology ◽  
Singular Homology

AbstractIn the category of soft topological spaces, a singular homology group is defined and the homotopic invariance of this group is proved [Internat. J. Engrg. Innovative Tech. (IJEIT) 3 (2013), no. 2, 292–299]. The first aim of this study is to define relative homology groups in the category of pairs of soft topological spaces. For these groups it is proved that the axioms of dimensional and exactness homological sequences hold true. The axiom of excision for singular homology groups is also proved.

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.

scholarly journals TRIPLY-GRADED LINK HOMOLOGY AND HOCHSCHILD HOMOLOGY OF SOERGEL BIMODULES

2007 ◽  
Vol 18 (08) ◽  
pp. 869-885 ◽  
Author(s):  
MIKHAIL KHOVANOV
Keyword(s):  
Group Action ◽  
Braid Group ◽  
Polynomial Algebra ◽  
Homology Theory ◽  
Hochschild Homology ◽  
Polynomial Algebras ◽  
Homology Groups ◽  
Link Homology ◽  
Soergel Bimodules ◽  
Braid Group Action

We consider a class of bimodules over polynomial algebras which were originally introduced by Soergel in relation to the Kazhdan–Lusztig theory, and which describe a direct summand of the category of Harish–Chandra modules for sl(n). Rouquier used Soergel bimodules to construct a braid group action on the homotopy category of complexes of modules over a polynomial algebra. We apply Hochschild homology to Rouquier's complexes and produce triply-graded homology groups associated to a braid. These groups turn out to be isomorphic to the groups previously defined by Lev Rozansky and the author, which depend, up to isomorphism and overall shift, only on the closure of the braid. Consequently, our construction produces a homology theory for links.

HOMOLOGY THEORIES FOR COMPLEXES BASED ON FLATS

2019 ◽  
Vol 63 (1) ◽  
pp. 27-44 ◽  
Author(s):  
LI LIANG
Keyword(s):  
Homology Theory ◽  
Relative Homology ◽  
Gorenstein Flat ◽  
Associative Rings

AbstractIn this paper, we introduce and study the Gorenstein relative homology theory for unbounded complexes of modules over arbitrary associative rings, which is defined using special Gorenstein flat precovers. We compare the Gorenstein relative homology to the Tate/unbounded homology and get some results that improve the known ones.

scholarly journals On Path Homology of Vertex Colored (Di)Graphs

Symmetry ◽  
2020 ◽  
Vol 12 (6) ◽  
pp. 965
Author(s):  
Yuri V. Muranov ◽  
Anna Szczepkowska
Keyword(s):  
Spectral Sequence ◽  
Homology Theory ◽  
Homotopy Invariance ◽  
Homology Groups ◽  
Basic Properties

In this paper, we construct the colored-path homology theory in the category of vertex colored (di)graphs and describe its basic properties. Our construction is based on the path homology theory of digraphs that was introduced in the papers of Grigoryan, Muranov, and Shing-Tung Yau and stems from the notion of the path complex. Any graph naturally gives rise to a path complex in which for a given set of vertices, paths go along the edges of the graph. We define path complexes of vertex colored (di)graphs using the natural restrictions that are given by coloring. Thus, we obtain a new collection of colored-path homology theories. We introduce the notion of colored homotopy and prove functoriality as well as homotopy invariance of homology groups. For any colored digraph, we construct the spectral sequence of colored-path homology groups which gives the effective method of computations in the general case since any (di)graph can be equipped with various colorings. We provide a lot of examples to illustrate our results as well as methods of computations. We introduce the notion of homotopy and prove functoriality and homotopy invariance of introduced vertexed colored-path homology groups. For any colored digraph, we construct the spectral sequence of path homology groups which gives the effective method of computations in the constructed theory. We provide a lot of examples to illustrate obtained results as well as methods of computations.

Skein Homology

1998 ◽  
Vol 41 (2) ◽  
pp. 140-144 ◽  
Author(s):  
Doug Bullock ◽  
Charles Frohman ◽  
Joanna Kania-Bartoszyńska
Keyword(s):  
Homology Theory ◽  
Homology Groups ◽  
Skein Module ◽  
New Class ◽  
Kauffman Bracket

AbstractA new class of homology groups associated to a 3-manifold is defined. The theories measure the syzygies between skein relations in a skein module. We investigate some of the properties of the homology theory associated to the Kauffman bracket.

A Homological characterization of certain Abelian groups

1961 ◽  
Vol 57 (2) ◽  
pp. 256-264 ◽  
Author(s):  
A. J. Douglas
Keyword(s):  
Abelian Group ◽  
Homology Group ◽  
Abelian Groups ◽  
Unit Element ◽  
Homology Theory ◽  
Homology Groups ◽  
Complete Answer ◽  
Homological Characterization

Let G be a monoid; that is to say, G is a set such that with each pair σ, τ of elements of G there is associated a further element of G called the ‘product’ of σ and τ and written as στ. In addition it is required that multiplication be associative and that G shall have a unit element. The so-called ‘Homology Theory’† associates with each left G-module A and each integer n (n ≥ 0) an additive Abelian group Hn (G, A), called the nth homology group of G with coefficients in A. It is natural to ask what can be said about G if all the homology groups of G after the pth vanish identically in A. In this paper we give a complete answer to this question in the case when G is an Abelian group. Before describing the main result, however, it will be convenient to define what we shall call the homology type of G. We write Hn(G, A) ≡ 0 if Hn(G, A) = 0 for all left G-modules A.

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