Singular Homology Theory Research Articles

The homology digraph of a preordered space

This paper studies a notion of directed homology for preordered spaces, called the homology digraph. We show that the homology digraph is a directed homotopy invariant and establish variants of the main results of ordinary singular homology theory for the homology digraph. In particular, we prove a Künneth formula, which enables one to compute the homology digraph of a product of preordered spaces from the homology digraphs of the components.

Singular cell homology

In the paper we generalize the singular cubical homology theory of digraphs and introduce a collection of cell singular homologies that are parametrized by a natural parameter of the category of digraphs. The natural parameter corresponds to the “length” of a directed line digraph that corresponds to the unit segment in algebraic topology. The introduced earlier singular cubical homology is a particular case of this collection that corresponds to the line digraph with one arrow. We describe relations between these theories for different parameters given by a natural number and study their basic properties. In particular, we prove the functoriality and homotopy invariance of these theories and compare the theory with the path homology theory that was introduced in our previous papers. Then we describe the transfer of the cell singular homology theories to the categories of graphs, quivers, and multigraphs.

Homological Algebra and Its Application: A Descriptive Study

Algebra has been used to define and answer issues in almost every field of mathematics, science, and engineering. Homological algebra depends largely on computable algebraic invariants to categorise diverse mathematical structures, such as topological, geometrical, arithmetical, and algebraic (up to certain equivalences). String theory and quantum theory, in particular, have shown it to be of crucial importance in addressing difficult physics questions. Geometric, topological and algebraic algebraic techniques to the study of homology are to be introduced in this research. Homology theory in abelian categories and a category theory are covered. the n-fold extension functors EXTn (-,-) , the torsion functors TORn (-,-), Algebraic geometry, derived functor theory, simplicial and singular homology theory, group co-homology theory, the sheaf theory, the sheaf co-homology, and the l-adic co-homology, as well as a demonstration of its applicability in representation theory.

Digital homotopy relations and digital homology theories

In this paper we prove results relating to two homotopy relations and four homology theories developed in the topology of digital images.<br /><br />We introduce a new type of homotopy relation for digitally continuous functions which we call ``strong homotopy.'' Both digital homotopy and strong homotopy are natural digitizations of classical topological homotopy: the difference between them is analogous to the difference between digital 4-adjacency and 8-adjacency in the plane.<br /><br />We also consider four different digital homology theories: a simplicial homology theory by Arslan et al which is the homology of the clique complex, a singular simplicial homology theory by D. W. Lee, a cubical homology theory by Jamil and Ali, and a new kind of cubical homology for digital images with $c_1$-adjacency which is easily computed, and generalizes a construction by Karaca \& Ege. We show that the two simplicial homology theories are isomorphic to each other, but distinct from the two cubical theories.<br /><br />We also show that homotopic maps have the same induced homomorphisms in the cubical homology theory, and strong homotopic maps additionally have the same induced homomorphisms in the simplicial theory.

Digital Hurewicz theorem and digital homology theory

In this paper, we develop homology groups for digital images based on cubical singular homology theory for topological spaces. Using this homology, we present digital Hurewicz theorem for the fundamental group of digital images. We also show that the homology functors developed in this paper satisfy properties that resemble the Eilenberg-Steenrod axioms of homology theory, in particular, the homotopy and the excision axioms. We finally define axioms of digital homology theory.

On the axiomatic systems of singular cohomology theory

On the category of pairs of topological spaces having a homotopy type of CW complexes the singular (co)homology theory was axiomatically studied by J. Milnor [8]. In particular, Milnor gave additivity axiom for a (co)homology theory and proved that any additive (co)homology theory on the given category is isomophic to the singular (co)homology. On the other hand, the singular homology is a homology with compact support [3]. In the paper [6], L. Mdzinarishvili proposed partially compact support property for a cohomology theory and gave another axiomatic characterization of the singular cohomology theory. In this paper, we will give additional different axiomatic characterizations of the singular cohomology theory. Moreover, we will study connections of the mentioned axiomatic systems (cf. [2]).

The uniqueness theorem for cohomologies on the category of polyhedral pairs

Let X be a topological space and F={Fα} be a direct system of all compact subsets Fα of X, directed by inclusions. For any homology theory H∗ the groups {H∗(Fα)∣Fα⊂X} constitute a direct system, and the maps H∗(Fα)→H∗(X) define a homomorphism i∗:lim⟶H∗(Fα)→H∗(X).As is known (Theorem 4.4.6, Spanier, 1966), for the singular homology, the homomorphism i∗ is an isomorphism (1)i∗:lim⟶H∗s(Fα)⟶∼H∗s(X).Using the isomorphism (1), it is proved that for the homologies having compact support H there is the uniqueness theorem on the category of polyhedral pairs (Theorem 4.8.14, Spanier, 1966).Since the singular homology theory is a homology theory with compact supports, the uniqueness theorem connects all homology theories having compact supports with the singular homology theory.Let H∗ be a cohomology theory. The groups {H∗(Fα)∣Fα⊂X} constitute an inverse system, and the maps H∗(X)→H∗(Fα) define a homomorphism i∗:H∗(X)→lim⟵H∗(Fα).Since the homology functor does not commute with inverse limits, it is not true that the singular cohomology of a space is isomorphic to the inverse limit of the singular cohomology of its compact subsets (that is, there is no general cohomology analogue of Theorem 4.4.6, Spanier, 1966).In the present work, it will be shown that there is such connection for a singular cohomology. Namely, there exists a finite exact sequence (2)0⟶lim⟵(2n−3)Hs1(Fα,G)⟶⋯⟶lim⟵(1)Hsn−1(Fα,G)⟶Hsn(X,G)⟶lim⟵Hsn(Fα,G)⟶lim⟵(2)Hsn−1(Fα,G)⟶⋯⟶lim⟵(2n−2)Hs1(Fα,G)⟶0.The terms the Alexander cohomology with compact supports and the singular cohomology with compact supports used in the works (Spanier, 1966; Mdzinarishvili, 1984) do not refer to our problem. Therefore, cohomology theory, in particular the singular cohomology, for which there is a finite exact sequence (2), is called a cohomology with partially compact supports.In the present work, using a finite exact sequence (2), it is proved the uniqueness theorem for a cohomology having partially compact supports on the category of polyhedral pairs. Hence, the uniqueness theorem connects all cohomology theories with partially compact supports with the singular cohomology theory.

Singular homology theory in the category of soft topological spaces

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.

Milnor–Thurston homology groups of the Warsaw Circle

Milnor–Thurston homology theory is a construction of homology theory that is based on measures. It is known to be equivalent to singular homology theory in case of manifolds and complexes. Its behaviour for non-tame spaces is still unknown. This paper provides results in this direction. We prove that Milnor–Thurston homology groups for the Warsaw Circle are trivial except for the zeroth homology group which is uncountable-dimensional. Additionally, we prove that the zeroth homology group is non-Hausdorff for this space with respect a natural topology that was proposed by Berlanga.

Singular homology of arithmetic schemes

We construct a singular homology theory on the category of schemes of finite type over a Dedekind domain and verify several basic properties. For arithmetic schemes we construct a reciprocity isomorphism between the integral singular homology in degree zero and the abelianized modified tame fundamental group.

Intersections and Translative Integral Formulas for Boundaries of Convex Bodies

Let K,L IR n be two convex bodies with non-empty interiors and with boundaries @K, @L, and let denote the Euler characteristic as defined in singular homology theory. We prove two translative integral formulas involving boundaries of convex bodies. It is shown that the integrals of the functions t 7! (@K (@L + t)) and t 7! (@K (L + t)), t 2 IR n , with respect to an n- dimensional Haar measure of IR n can be expressed in terms of certain mixed volumes of K and L. In the particular case where K and L are outer parallel bodies of convex bodies at distance r > 0, the result will be deduced from a recent (local) translative integral formula for sets with positive reach. The general case follows from this and from the following (global) topological result. Let Kr,Lr denote the outer parallel bodies of K,L at distance r 0. Establishing a conjecture of Firey (1978), we show that the homotopy type of @Kr @Lr and @Kr Lr, respectively, is independent of r 0 if K L 6 ; and if @K and @L intersect almost transversally. As an immediate consequence of our translative integral formulas, we obtain a proof for two kinematic formulas which have also been conjectured by Firey.

Singular homology and class field theory of varieties over finite fields

Article Singular homology and class field theory of varieties over finite fields was published on October 20, 2000 in the journal Journal für die reine und angewandte Mathematik (volume 2000, issue 527).

On the (non)-coincidence of Milnor–Thurston homology theory with singular homology theory

The paper investigates a homology theory based on the ideas of Milnor and Thurston that by considering measures on the set of all singular simplices one should get alternate possibilities for describing the cycles of classical homology theory. It suggests slight changes to Milnor’s and Thurston’s original definitions (giving differences for wild topological spaces only) which ensure that their homology theory is well-defined on all topological spaces. It further proves that Milnor-Thurston homology theory gives the same homology groups as the singular homology theory with real coefficients for all triangulable spaces. An example showing that the coincidence between these both homology theories does not hold for all topological spaces is also included.

An Extension of Baum-Fulton-MacPherson's Riemann-Roch Theorem for Singular Varieties

In [1] Baum, Fulton and MacPherson extend the Grothendieck-RiemannRoch theorem (abbr. GRR) for non-singular projective varieties to (possibly singular) quasi-projective varieties over any field and proper morphisms. More precisely, in [1, Chapter 1] they prove the Riemann-Roch (abbr. RR) in the case of (possibly singular) complex projective varieties with the ordinary singular homology theory with rational coefficients, by deformation to the normal bundle, and in [1, Chapter 2], in more general, they prove the RR in the case of (possibly singular) quasi-projective varieties over any field and proper morphisms with the Chow homology theory with rational coefficeints [3], by the Grassmannian graph construction. If the field is the complex numbers C, then the above Chow homology theory can be replaced by the singular homology theory (and Borel-Moore homology theory for non-compact varieties). Thus [1, Chap. 2] generalizes [1, Chap. 1].

On the cohomology of ring extensions

In this article we are primarily concerned with the relativized cohomology theory of ring extensions, as initiated by G. Hochschild [ 141. Our main objective is to lay the foundation for a general vanishing theory of relativized extension functors. The significance of such a theory rests on the unification it provides for the various classical results from the cohomology theories of groups, associative algebras, and Lie algebras. The approach chosen here heavily emphasizes locally nilpotent operators as an underlying principle for vanishing and isomorphism theorems. Although in algebraic homology theory these concepts have appeared only recently (cf. [5]), they occur implicitly in singular homology theory of topological spaces. The conventional proof of the excision theorem, for instance, resorts, by means of barycentric subdivision, to the construction of a locally nilpotent operator that is chain homotopic to the identity. In Section 1 we study the question to what an extent properties of operators are inherited by resolutions. It is shown that resolutions defined by injective envelopes and projective covers are suitable tools for ordinary homology. In the framework of relative homological algebra of a given ring extension R:S the availability of standard resolutions allows the application of more computational techniques. We show that the commutator algebra of the centralizer C,(S) of S in R operates on these resolutions in a natural fashion. This result in conjunction with a certain homotopy is subsequently utilized to show that the associated action on the spaces Ext;,, S) (M, N) is trivial. The succeeding section combines these techniques with basic facts concerning locally finite operators to establish various vanishing theorems for extension functors. The third section deals with the study of the cohomology theory of Frobenius extensions. We address questions emerging from recent developments in the cohomology theories of finite dimensional modular [9] and finite dimensional restricted Lie algebras [7-93. Our results are closely

Book Review: Singular homology theory

Book Review: Singular homology theory

The topology of complex projective varieties after S. Lefschetz

AFTER THE topology of complex algebraic curves, i.e. the genus of Riemannian surfaces, had been understood mathematicians like Picard [12] and PoincarC [ 12a] went on to the next dimension and began to investigate the topology of complex algebraic surfaces. From 1915 on Lefschetz continued their work and extended it to higher dimensional varieties. In 1924 he published his famous exposition [L] of this work. When it was written knowledge of topology was still primitive and Lefschetz “made use most uncritically of early topology g la PoincarC and even of his own later developments”?. This makes it nowadays rather difficult to understand the topological parts of [L] properly. But that is not the only difficulty: Implicitly Lefschetz quite often appeals to geometric intuition where we would like to see a more precise argument. Thus there is some temptation to discard Lefschetz’s original “proofs” and adopt instead the more recent methods which have been employed to obtain many of his results, using Hodge’s theory of harmonic differential forms or Morse theory or sheaf theory and spectral sequences. But none of these very elegant methods yields Lefschetz’s full geometric insight, e.g. they do not show us the famous “vanishing cycles”. The first attempt to rewrite the topological part of [L] using modern singular homology theory was made more than twenty years ago by Wallace [16]. But the details of his presentation are too complicated to popularize Lefschetz’s original methods. Wallace leaves the realm of algebraic geometry far too early when he makes Lefschetz’s intuitive arguments precise. Furthermore he does not give a complete picture of Lefschetz’s achievements. In the following I make a new attempt to present Lefschetz’s almost sixty year old investigations rigorously but as geometrically as he did in [L]. For topologists Lefschetz is usually interesting for the work he did in pure topology after he had completed [L]. But [Ll has at least “a unique historical interest in being almost the first account of the topology of a construct of importance in general mathematics which is not trivial” (Hedge). We may furthermore speculate how much of the contributions of Poincare, Lefschetz and others to algebraic topology we owe to the difficulties they encountered with the topology of algebraic varieties. The necessary prerequisites in algebraic geometry can be found in the first two chapters of Shafarevich’s book[l3]. The main tool from differential topology is Ehresmann’s fibration theorem, which for the convenience of the reader is stated in 3.0. (Strangely enough this theorem is not included in the standard textbooks.) As far as homology theory is concerned a textbook like Dold’s[6] will amply suffice. Furthermore some basic facts about the fundamental group and the homotopy lifting theorem for fibre bundles will be used.

Singul�re definition der �quivarianten Bredon homologie

Let G be a discrete group,o(G) the orbit category of G and M:o(G)→a a covariant (contravariant) functor to abelian groups. We define a singular equivariant homology theory H*(X;M) (resp. H*(X;M)) which satisfies a dimension axiom, in the sense of Bredon (Lecture notes 34). It turns out, that all fundamental properties of these theories directly follow by naturality from the analogous theorems in the classical non equivariant case.

On a Homology Theory Associated to Foliations

In this note, we associate a foliated singular homology theory to a foliation on a compact manifold X and construct a spectral sequence which relates the foliated homology to the ordinary homology of X. Since the foliated homology is so constructed as to be related closely to a certain topological behavior of the foliation, we may expect that further study of the spectral sequence reveals some information on topology of foliation. The study and applications will be given in a sequel.

Flat and coherent functors

The main objective of this paper is to characterize flat group valued functors. We obtain the following theorem, announced in [;7: Let X be a small additive category with dual X0 and S an object in [x0, AB], the category of all additive functors from X to the category Ab of Abelian groups. Then S is flat, i.e., the functor S @r : [X, AB] + AB is exact if and only if the fiber X/S of the Yoneda embedding X --+ [x0, AB] over S is filtered from above, or if and only if S is a filtered direct limit of representable functors. There are several other equivalent statements, and it is, mutatis mutandis, enough to aSsume X preadditive. A similar theorem has been obtained by B. Stenstrom in [II]. He proves that a functor is flat if and only if it is a filtered direct limit of projective (instead of representable) functors. For Abelian X the result was obtained by J. Fisher [5J; in this particular case “flat” means “left exact,” and a short proof is possible. Our result is a generalization of the well-known characterization of flat modules by means of generators and relations, and has applications in the study of the exactness of the direct limit functor [7], and in the singular homology theory of sheaves [S]. Using the above characterization of flat functora we show in analogy to the results of S. U. Chase [4] on coherent rings that the category [X0, AB] is locally coherent, i.e., has a family of coherent generators, if and only if