Cochain Complex Research Articles

A note on module homotopy and chain homotopy

The Eckmann-Hilton (injective) homotopy category of modules over a ring Λ \Lambda is equivalent to a certain full subcategory of the cochain homotopy category of cochain complexes of modules over Λ \Lambda .

ON THE COCHAIN COMPLEX OF TOPOLOGICAL SPACES

Let be a topological space and the cochain complex with coefficients in .In this paper it is shown that one can define on the structure of algebras over an -operad, and the -construction is carried out. The -construction is an -algebra that makes it possible to define an -structure on the cochain complex of loop spaces.Bibliography: 7 titles.

Group Extensions and Cohomology for Locally Compact Groups. III

We shall define and develop the properties of cohomology groups ${H^n}(G,A)$ which can be associated to a pair (G, A) where G is a separable locally compact group operating as a topological transformation group of automorphisms on the polonais abelian group A. This work extends the results in [29] and [30], and these groups are to be viewed as analogues of the Eilenberg-Mac Lane groups for discrete G and A. Our cohomology groups in dimension one are classes of continuous crossed homomorphisms, and in dimension two classify topological group extensions of G by A. We characterize our cohomology groups in all dimensions axiomatically, and show that two different cochain complexes can be used to construct them. We define induced modules and prove a version of Shapiro’s lemma which includes as a special case the Mackey imprimitivity theorem. We show that the abelian groups ${H^n}(G,A)$ are themselves topological groups in a natural way and we investigate this additional structure.

Group extensions and cohomology for locally compact groups. III

We shall define and develop the properties of cohomology groups H n ( G , A ) {H^n}(G,A) which can be associated to a pair (G, A) where G is a separable locally compact group operating as a topological transformation group of automorphisms on the polonais abelian group A. This work extends the results in [29] and [30], and these groups are to be viewed as analogues of the Eilenberg-Mac Lane groups for discrete G and A. Our cohomology groups in dimension one are classes of continuous crossed homomorphisms, and in dimension two classify topological group extensions of G by A. We characterize our cohomology groups in all dimensions axiomatically, and show that two different cochain complexes can be used to construct them. We define induced modules and prove a version of Shapiro’s lemma which includes as a special case the Mackey imprimitivity theorem. We show that the abelian groups H n ( G , A ) {H^n}(G,A) are themselves topological groups in a natural way and we investigate this additional structure.

Curvature groups of a hypersurface

A cochain complex associated with the vector 1 1 -form determined by the first and second fundamental tensors of a hypersurface M M in E n + 1 {E^{n + 1}} is introduced. Its cohomology groups H p ( M ) {H^p}(M) , called curvature groups, are isomorphic with the cohomology groups of M M with coefficients in a subsheaf S R {\mathcal {S}_R} of the sheaf S \mathcal {S} of closed vector fields on M M . If M M is a minimal variety, the same conclusion is valid with S R {\mathcal {S}_R} replaced by a sheaf of harmonic vector fields. If the Ricci tensor is nondegenerate the H p ( M ) {H^p}(M) vanish. If S R ≠ ∅ {\mathcal {S}_R} \ne \emptyset , and there are no parallel vector fields, locally, the H p ( M ) {H^p}(M) are isomorphic with the corresponding de Rham groups.

The cohomology ring of a smooth manifold

A fundamental result relating the topology of a smooth manifold and its global differential geometry is the theorem of De Rham. Let Y(X, R) denote the exterior algebra (over the reals R) of smooth differential forms on X, and let &(X, R) denote its derived cohomology algebra. The De Rham theorem asserts that there is an algebra isomorphism of &(X, R) with the singular cohomology algebra (cup product) H(X, R) of X. A generalization of this theorem to handle other coefficient domains is due to C. B. Allendoerfer and J. Eells, Jr. in [1]. For an arbitrary integral subdomain A of R they define a cochain complex (Y(X, A)-the complex of A pairs of forms on Xand prove that the derived cohomology module &(X, A) of ;(YX, A) is canonically isomorphic to the Cech cohomology module H(X, A) with values in A. This paper is a natural extension of the Allendoerfer-Eells paper. Our concern is to define a ring structure for &(X, Z), where Z denotes the integers, so that the canonical (module) isomorphism with H(X, Z) preserves products. The key is to define a certain cochain map tY(X, Z) 0 (S(Y, Z) -? Y(X x Y, Z), where X and Y are any smooth manifolds. This defines an exterior cup product SD(X, Z) 0 St Y, Z)-> I& (Xx Y, Z) which sends 6 0 Xj E Sv(X, Z) 0 q( Y, Z) to Xxr -E SP+q(Xx Y, Z). Let hl: &T(X, Z)H(X, Z), h2: I(Y, Z) --fI(YZ) and h: (X x Y, Z) -> H(X x Y, Z) be the canonical isomorphisms. Then a sheaf argument shows that h(e xq) = hl(e) x h2(q) where hl(e) x h2G0) is the exterior cup product (or cartesian product) of hl(e) and h2(-) in the Cech theory. We define the interior cup product for &~(X, Z) via the homomorphism A*: &)(Xx X, Z)-. &(X, Z) induced by the diagonal map A: X --X x X and thus obtain a cohomology algebra St(X, Z) isomorphic with the cohomology algebra IH(X, Z). Finally we remark that the same method applies to arbitrary coefficient rings A in R to define an algebra &(X, A) algebra isomorphic with Hi(X, A). The paper is organized as follows. The first section is devoted to a restatement of some of the definitions of [1]. In ?2 we discuss the general problem of introducing a product in &(X, Z). In ?3 we prove a lemma regarding general position of chains and singular sets which is needed in a later section. ?4 is devoted to proving the theorems stated in ?2. In ?5 we construct the exterior and interior cup products, and in ?6 prove that the isomorphism extends to products. In ?7 we make some remarks concerning products when X is given a triangulation.

Holomorphic Cohomology of Complex Analytic Linear Groups

Let G be a complex analytic group, and let A be the representation space of a finite-dimensional complex analytic representation of G. We consider the cohomology for G in A, such as would be obtained in the usual way from the complex of holomorphic cochains for G in A. Actually, we shall use a more conceptual categorical definition, which is equivalent to the explicit one by cochains. In the context of finite-dimensional representation theory, nothing substantial is lost by assuming that G is a linear group. Under this assumption, it is the main purpose of this paper to relate the holomorphic cohomology of G to Lie algebra cohomology, and to the rational cohomology, in the sense of [1], of algebraic hulls of G. This is accomplished by using the known structure theory for complex analytic linear groups in combination with certain easily established results concerning the cohomology of semidirect products. The main results are Theorem 4.1 (whose hypothesis is always satisfied by a complex analytic linear group) and Theorems 5.1 and 5.2. These last two theorems show that the usual abundantly used connections between complex analytic representations of complex analytic groups and rational representations of algebraic groups extend fully to the superstructure of cohomology.

On the Cohomology of Compact Homogeneous Spaces of Nilpotent Lie Groups

As part of the general problem of investigating the topological structure of homogeneous spaces of non-compact Lie groups, A. Malcev [7]2 obtained the following results. Any homogeneous space of a connected nilpotent Lie group is the topological product of an euclidean space and a compact space which is itself a homogeneous space of a certain connected nilpotent Lie group. If the homogeneous space is compact, it can be expressed in the form 9I = 3/T where 5 is a simply connected nilpotent Lie group and Z a discrete subgroup. Recently, Y. Matsushima [8] has proved that the cohomology groups H*(9) (with real coefficients) of the homogeneous space 9I are isomorphic with the cohomology groups H* (g) of the Lie algebra g of 5 for dimensions r = 1 and 2. The first and main purpose of the present paper is to establish the isomorphism of these cohomology groups for every dimension r. Indeed, we shall prove Theorem 1 which gives a canonical isomorphism of the cohomology algebras H*(9)) and H*(g). As corollaries we obtain a few results on the Euler characteristic and Betti numbers of compact homogeneous spaces of nilpotent Lie groups, which are similar to known results concerning homogeneous spaces of compact Lie groups. It also follows that a homogeneous space of a connected nilpotent Lie group is always orientable. For the homogeneous spaces of compact Lie groups we have the well known theory of invariant integrals of E. Cartan, namely, the cohomology of a homogeneous space of a compact Lie group can be obtained from the complex of invariant differential forms on it (for example, [4], Chapter I, Theorem 2.3). The second purpose of this paper is to show that, in the notation 9 = = / as above, the complex of invariant differential forms on 91 is isomorphic with the complex of invariant cochains of the Lie algebra g (Theorem 2). Then we shall give a very simple example to the effect that the theory of invariant integrals does no longer hold for compact homogeneous spaces of non-compact Lie groups. Such examples do not seem to have been known. For the proof of Theorem 1 we need some fundamental concepts and results in the cohomology theory of principal fiber bundles as well as the theory of differential sequences in the sense of Leray and Koszul. In ?2 we shall give those concepts which are necessary for our purpose and prove a lemma from which Theorem 1 can be easily derived. The formulation and proof of Theorem 1 to-

Effectiveness at the origin of the sum set of basic sets of polynomials

an equivalence relation only at one stage, namely, when cocycles are identified modulo coboundaries. Actually, we arrived at this definition of HP(X, A) while trying to set up the relative cohomology groups subject to the following two conditions. (i) HP(X, A) should be the cohomology group of a cochain complex associated with the pair (X, A). (ii) The use of the process of identification should be minimized. Condition (i) is motivated by the desire to take advantage of the technical convenience of the general theory of cochain complexes, while condition (ii) is concerned with conceptual simplicity.

Exact Homomorphism Sequences in Homology Theory

The developments of this paper stem from the attempts of one of the authors to deduce relations between homology groups of a complex and homology groups of a complex which is its image under a simplicial map. Certain relations were deduced (see [EP 1] and [EP 2]) which form an extension of the Mayer-Vietoris formulas concerning coverings of a complex by two complexes. In this paper these relations are formalized and are seen to be consequences of the existence of an exact homomorphism sequence (see Definition 3.1). The concept of an exact sequence seems to be due to Hurewicz [WH]. Its principal property is used in the presentation of the Mayer-Vietoris formulas by Alexandroff and Hopf [A-H, pp. 297-299]. It has been used most notably by Eilenberg and Steenrod (reference [E-S]) as one of a very simple system of axioms for homology theory. (See [C] for another use.) In one sense, this paper might have been written from the point of view of exploiting this axiom. Actually we have found the most flexible approach to be the consideration of the exact sequence of homology groups on a Mayer complex as a fundamental algebraic identity. We exploit this identity in two directions. First we obtain a number of duality theorems and second, we investigate chain mappings and some closely related topics on coverings. A considerable part of the paper is methodological in character in that known results are deduced as part of a general line of reasoning. In geometric applications involving Cech homology groups there are two procedures available. One might consider fundamental complexes for a compact metric space, in which case the fundamental algebraic identity could be used, or else one might set up a limiting process. We have used the latter method. Section 1 is concerned with preliminary remarks on notation. In Section 2 we define the term homomorphism sequence and recall the definition of an abstract complex according to Mayer [WM 1, 2, 4] distinguishing chain and cochain complexes notationally and defining homology and cohomology groups. In Section 3 we define the term exact sequence and establish the fundamental construction of the paper, Theorem 3.3. In Section 4, the limit of a direct system of homomorphism sequences is defined and direct limits of exact sequences are shown to be exact. Section 5 gives a summary of needed results on character theory. Compare with papers by Alexandroff [A] and Mayer [WM 4]. These results are used in Section 6 to establish algebraic duality in Mayer complexes. Compare with [WM 4]. The results of Section 5 are used again in Section 7, where limits of inverse systems of homomorphism sequences of compact groups are defined, to show that inverse limits of exact sequences of compact groups are compact. In Section 8, the theory of inverse limits is applied to prove exactness of the homomorphism sequence of homology groups relative to a compact coefficient

Curved $A_{\infty}$-algebras and Chern classes

We describe two constructions giving rise to curved $A_{\infty}$-algebras. The first consists of deforming $A_{\infty}$-algebras, while the second involves transferring curved dg structures that are deformations of (ordinary) dg structures along chain contractions. As an application of the second construction, given a vector bundle on a polyhedron $X$, we exhibit a curved $A_{\infty}$-structure on the complex of matrix-valued cochains of sufficiently fine triangulations of $X$. We use this structure as a motivation to develop a homotopy associative version of Chern-Weil theory.

Periodic symplectic cohomologies

Goodwillie \cite{Goodwillie} introduced a periodic cyclic homology group associated to a mixed complex. In this paper, we apply this construction to the symplectic cochain complex of a Liouville domain $M$ and obtain two periodic symplectic cohomology theories, denoted as $HP^*_{S^1}(M)$ and $HP^*_{S^1, \mathrm{loc}}(M)$. Our main result is that both cohomology theories are invariant under Liouville isomorphisms and there is a natural isomorphism $HP^*_{S^1, \mathrm{loc}}(M, \mathbb{Q}) \cong H^*(M, \mathbb{Q})((u))$, which can be seen as a localization theorem for $HP^*_{S^1, \mathrm{loc}}(M, \mathbb{Q})$.