THE FUNDAMENTALS AND RUDIMENTS OF HOMOLOGICAL ALGEBRA

Mathematics and Statistics, 2021

Polynomial: algebra is essential in commutative algebra since it can serve as a fundamental model for differentiation. For module differentials and Loday's differential commutative graded algebra, simplified homology for polynomial algebra was defined. In this article, the definitions of the simplicial, the cyclic, and the dihedral homology of pure algebra are presented. The definition of the simplicial and the cyclic homology is presented in the Algebra of Polynomials and Laurent's Polynomials. The long exact sequence of both cyclic homology and simplicial homology is presented. The Morita invariance property of cyclic homology was submitted.

Transactions of the American Mathematical Society, 1962

The mapping functor F plays a role in topology analogous to that played by Horn in group theory. We seek an analogue of the tensor product; this is provided by the reduced join /\; in fact, the reduced join and mapping functors are adjoint functors in the sense of Kan [13] because of the well-known relation F(XA Y, Z) m F(X, F(Y, Z)), valid for well-behaved spaces. Now if X is a space and E a spectrum, then E/\X is again a spectrum and we define the generalized homology groups Hn(X;E) = irn(E/\X). We prove that the generalized homology groups satisfy the Eilenberg-Steenrod axioms, except for the dimension axiom. With an appropriate notion of pairing of spectra, we can define cup-and cap-products. Using these we then prove an Alexander duality theorem. Moreover, we characterize the class of manifolds satisfying Poincaré duality for arbitrary spectra; it includes the n-manifolds of J. H. C. Whitehead [35] and of Milnor [19]. The results of this paper were announced in [32]. §2 is devoted to general preliminaries, and §3 to homology and homotopy properties of the reduced join. Most of the results of these sections are well-License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1962] GENERALIZED HOMOLOGY THEORIES 229 objects and maps of "Wo! the terms "free space" and "free map" will refer to objects and maps of "W. Let n be a positive integer. By "w-ad" we shall mean an «-ad (X; Xx, ■ • • , Xn-x\ #0) having the same homotopy type as some CW-wad

homology In this book our attention will be focused on compact, simply connected, dif-ferentiable 4-manifolds. The restriction to the simply connected case certainly rules out many interesting examples :indeed it is well-known that any finitely presented group can occur as the fundamental group of a 4-manifold.Furthermore the techniques we will develop in the body of the book are in reality rather insensitive to the fundamental group and much of our discussion can easily be generalized. The main issues however can be reached more quickly in the simply connected case. We shall see that for many purposes 4-manifolds with trivial fundamental group are of beguiling simplicitly but nevertheless the most basic questions about the differential topology of these manifolds lead us into new uncharted waters where the results described in this book serve at present,as isolated markers. After the fundamental group we have the homology and cohomology groups of a 4-manifold. For a closed oriented 4-manifold Poincare duality gives an isomorphism between homology and cohomology in complementary dimensions i and 4-i. So, when X is simply connected the first and third homology groups vanish and all the homological information is contained in H 2. The universal coefficient theorem for cohomology implies that when H 1 is zero H 2 is a free abelian group.In turn by Poincare duality the homology group is free. There are three concrete ways in which we can realize 2-dimensional ho-mology or cohomology classes on a 4-manifold and it is useful to be able to translate easily between them. The first is complex line bundles complex vector bundles of rank 1. On any space X a line bundle L is determined up to the bundle isomorphism by its Chern class and this sets up a bijection between the isomorphism classes of line bundles. The second realization is by smoothly embedded two-dimensional oriented surfaces in the manifold X. Such a surface carries a fundamental homology class ,given a line bundle L we can choose a general smooth section of the bundle whose zero set is a surface representing the homology class dual to c 1 (L) Third we have the de Rham representation of real cohomology classes by differential forms. Let X be a compact oriented simply connected four-manifold. The Poincare duality isomorphism between homology and cohomology is equivalent to a bi-linear form Q. This is the intersection form of the manifold. It is a unimodular symmetric form.Geometrically two oriented surfaces in X placed in general position will meet in a finite set of points. To each point we associate a sign ±1 according to the matching of the orientations in the isomorphism of the tangent bundles at that point. The intersection number is given by the total number of points counted with the signs. The pairing passes to homology to yield the form Q.Going over to cohomology the form translates into the cup product. Thus the form is an invariant of the oriented homotopy type of X. In terms of de Rham cohomology if w 1 and w 2 are closed 2-forms representing classes dual to the surfaces the intersection number is given by the integral X w 1 ∧ w 2 .

2008

In this article we extend some theorems in Homological Algebra. we show that if 0 → X n-(k+1) → X n-k → ... → X n-1 → X n → 0 is an exact sequence of zero sequence, then for every k ∈ N , there exist a natural homeomorphism ϕ k : (k+1) ). Also by using α-sequences, we define CH(A) = Im(A i A i+1 ) Ker(A i+1 A i+2 ) , and prove that, if T (A) is a additive exact functor, X is a α-complex, and if T is covariant in A, then for every n ∈ N , CH n (T (X)) ∼ = T (CH n (X)).

Mathematics and Statistics, 2022

This paper introduces a new idea in the unital involutive Banach algebras and its closed subset. This paper aims to study the cohomology theory of operator algebra. We will study the Banach algebra as an applied example of operator algebra, and the Banach algebra will be denoted by 𝒜𝒜. The definitions of cyclic, simplicial, and dihedral cohomology group of 𝒜𝒜 will be introduced. We presented the definition of ℬ-relative dihedral cohomology group that is given by:

Inventiones Mathematicae, 1993

The paper provides a homological algebraic foundation for semi-infinite cohomology.

Homology, Homotopy and Applications, 2005

It is proved that the homology and cohomology theories of groups and associative algebras are non-abelian derived functors of the cokernel and kernel groups of higher dimensions of their defining standard chain and cochain complexes respectively. The same results are also obtained for the relative (co)homology of groups, the mod q cohomology of groups and the cohomology of groups with operators. This allowed us to give an alternative approach to higher Hopf formulas for integral homology of groups. An axiomatic characterization of the relative cohomology of groups is given and higher relative (n + 1)-th cohomology of groups is described in terms of n-fold extensions.

Cladistics, 2012

Homology in cladistics is reviewed. The definition of important terms is explicated in historical context. Homology is not synonymous with synapomorphy: it includes symplesiomorphy, and Hennig clearly included both plesiomorphy and synapomorphy as types of homology. Homoplasy is error, in coding, and is analogous to residual error in simple regression. If parallelism and convergence are to be distinguished, homoplasy would be evidence of the former and analogy evidence of the latter. We discuss whether there is a difference between molecular homology and morphological homology, character state homology, nested homology (additive characters), and serial homology. We conclude by proposing a global definition of homology.

Proceedings of the American Mathematical Society, 1991

The effect on the cyclic homology of an algebra of adjoining a unit is calculated by means of an elementary analysis at the level of chains.

Applied Mathematical Sciences, 2019

In the present work, we increment to get another definition for Steenrod operations of the dihedral homology of commutative Hopf algebra. This appears by using the tensor product for the symmetric free resolution group, and the standard Hopf algebra resolution with the dihedral class. Furthermore, we fuse and handle several of operations.