The spectral sequence of an equivariant chain complex and homology with local coefficients (original) (raw)

The equivariant spectral sequence and cohomology with local coefficients

In his pioneering work from the late 1940s, J.H.C. Whitehead established the category of CW-complexes as the natural framework for much of homotopy theory. A key role in this theory is played by the cellular chain complex of the universal cover of a connected CW-complex, which in turn is tightly connected to (co-)homology with local coefficients. In [8], we revisit these classical topics, drawing much of the motivation from recent work on the topology of complements of complex hyperplane arrangements, and the study of cohomology jumping loci.

A Spectral Sequence for the Homology of a Finite

2016

In 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.

A spectral sequence for the homology of a finite algebraic delooping

Journal of K-theory: K-theory and its Applications to Algebra, Geometry, and Topology, 2014

The realization of polynomial algebras as cohomology rings

Pacific Journal of Mathematics, 1974

We construct, for certain choices of a group G, a prime p, and a positive integer n, a space X{G, p, n) whose cohomology ring mod p is a polynomial algebra, and we classify the polynomial algebras which can be realized as cohomology rings by this construction. Let Z p denote the ring of p-adic integers. From Sullivan's work on completions [15] it follows that the Eilenberg-MacLane space K(Z%, 2) is the p-profinite completion of K(Z n , 2), and that as a consequence of the p-analogue of [15, 3.9] we have H*(K(Zl 2); Z v) = Z p [x lf x 2 , , xΛ where deg x { = 2. Now if G is a subgroup of GL(n, Z p) and finite, we have an action of G on the space K(Z n p , 2) which passes to its cohomology ring, and we define X(G,p,n) = K(Z;,2)x G EG where EG is the total space of a universal bundle for G. PROPOSITION. If p does not divide the order of G, then H*(X(G, p } n); Z p) is the subalgebra of invariants of H*(K)(Z n p , 2); Z p) under the action of G. Obviously the conclusions of this proposition apply as well with coefficients in the prime field F p or in the field Q p of p-adic numbers. For the sake of completeness we sketch a proof. Proof. From [5, Th. 3.1] and [8] it follows that the cohomology of X(G, p, n) is given by Ext Zp[σ) (C*(EG)), C*(K(Z* P , 2)), where we let Z P (G) denote the group ring over Z p and C* and C* denote singular chains with coefficients in Z p. The Eilenberg-Moore spectral sequence associated with this Ext has E 2 term determined by El>* = Extί p(β) (Z p , H°(K(Zl 2); Z p))

On a spectral sequence for twisted cohomologies

Chinese Annals of Mathematics, Series B, 2014

Let (Ω * (M ), d) be the de Rham cochain complex for a smooth compact closed manifolds M of dimension n. For an odd-degree closed form H, there are a twisted de Rham cochain complex (Ω * (M ), d + H∧) and its associated twisted de Rham cohomology H * (M, H). We show that there exists a spectral sequence {E p,q r , dr} derived from the filtration Fp(Ω * (M )) = i≥p Ω i (M ) of Ω * (M ), which converges to the twisted de Rham cohomology H * (M, . We also show that the differentials in the spectral sequence can be given in terms of cup products and specific elements of Massey products as well, which generalizes a result of Atiyah and Segal. Some results about the indeterminacy of differentials are also given in this paper.

An Introduction to the Cohomology of Groups

H n (G, M) where n = 0, 1, 2, 3,. . ., called the nth homology and cohomology of G with coefficients in M. To understand this we need to know what a representation of G is. It is the same thing as ZG-module, but for this we need to know what the group ring ZG is, so some preparation is required. The homology and cohomology groups may be defined topologically and also algebraically. We will not do much with the topological definition, but to say something about it consider the following result: THEOREM (Hurewicz 1936). Let X be a path-connected space with π n X = 0 for all n ≥ 2 (such X is called 'aspherical'). Then X is determined up to homotopy by π 1 (x). If G = π 1 (X) for some aspherical space X we call X an Eilenberg-MacLane space K(G, 1), or (if the group is discrete) the classifying space BG. (It classifies principal G-bundles, whatever they are.) If an aspherical space X is locally path connected the universal cover˜X is contractible and X = ˜ X/G. Also H n (X) and H n (X) depend only on π 1 (X). If G = π 1 (X) we may thus define H n (G, Z) = H n (X) and H n (G, Z) = H n (X) and because X is determined up to homotopy equivalence the definition does not depend on X. As an example we could take X to be d loops joined together at a point. Then π 1 (X) = F d is free on d generators and π n (X) = 0 for n ≥ 2. Thus according to the above definition H n (F d , Z) = Z if n = 0 Z d if n = 1 0 otherwise. Also, the universal cover of X is the tree on which F d acts freely, and it is contractible. The theorem of Hurewicz tells us what the group cohomology is if there happens to be an aspherical space with the right fundamental group, but it does not say that there always is such a space.

Covering homology

Advances in Mathematics, 2010

We introduce the notion of covering homology of a commutative S-algebra with respect to certain families of coverings of topological spaces. The construction of covering homology is extracted from Bökstedt, Hsiang and Madsen's topological cyclic homology. In fact covering homology with respect to the family of orientation preserving isogenies of the circle is equal to topological cyclic homology. Our basic tool for the analysis of covering homology is a cofibration sequence involving homotopy orbits and a restriction map similar to the restriction map used in Bökstedt, Hsiang and Madsen's construction of topological cyclic homology. Covering homology with respect to families of isogenies of a torus is constructed from iterated topological Hochschild homology. It receives a trace map from iterated algebraic K-theory and there is a hope that the rich structure, and the calculability of covering homology will make it useful in the exploration of J. Rognes' "red shift conjecture".

K T ] 1 J ul 2 00 3 Hopf-cyclic homology and cohomology with coefficients

2003

Following the idea of an invariant differential complex, we construct general-type cyclic modules that provide the common denominator of known cyclic theories. The cyclicity of these modules is governed by Hopfalgebraic structures. We prove that the existence of a cyclic operator forces a modification of the Yetter-Drinfeld compatibility condition leading to the concept of a stable anti-Yetter-Drinfeld module. This module plays the role of the space of coefficients in the thus obtained cyclic cohomology of module algebras and coalgebras, and the cyclic homology and cohomology of comodule algebras. Along the lines of Connes and Moscovici, we show that there is a pairing between the cyclic cohomology of a module coalgebra acting on a module algebra and closed 0-cocycles on the latter. The pairing takes values in the usual cyclic cohomology of the algebra. Similarly, we argue that there is an analogous pairing between closed 0-cocycles of a module coalgebra and the cyclic cohomology of...

The spectral sequence of an equivariant chain complex and homology with local coefficients (original) (raw)

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