Sequence For Cohomology Research Articles

Cohomological Stabilization of Maximal Ideal Spaces in Banach Algebras

We prove that for a complex commutative Banach algebra A with finite dimensional character space X, the Bass stable rank condition sr A ≤ n is equivalent to a certain stabilization property for Čech cohomology sequences of pairs ( X, Z), where Z is the hull of an arbitrary closed ideal. In particular, the k-cohomology groups of X, Z and ( X, Z) vanish when k ≥ 2 n + 1. Using this result and Michael′s continuous selection theory we improve previous estimations for sr C ( Y, A), where C ( Y, A) denotes the algebra of continuous functions from a finite dimensional compact space Y into A.

A coarse Mayer–Vietoris principle

In [1], [4], and [6] the authors have studied index problems associated with the ‘coarse geometry’ of a metric space, which typically might be a complete noncompact Riemannian manifold or a group equipped with a word metric. The second author has introduced a cohomology theory, coarse cohomology, which is functorial on the category of metric spaces and coarse maps, and which can be computed in many examples. Associated to such a metric space there is also a C*-algebra generated by locally compact operators with finite propagation. In this note we will show that for suitable decompositions of a metric space there are Mayer–Vietoris sequences both in coarse cohomology and in the K-theory of the C*-algebra. As an application we shall calculate the K-theory of the C*-algebra associated to a metric cone. The result is consistent with the calculation of the coarse cohomology of the cone, and with a ‘coarse’ version of the Baum–Connes conjecture.

CONNECTION HOMOLOGY AND COHOMOLOGY BETWEEN SETS. ENCLOSURE HOMOLOGY AND COHOMOLOGY OF A CLOSED SET

The notions of connection homology and cohomology between complementary subsets of a topological space are defined, using passage to the limit with respect to boundary-open sets, i.e., complements of pairs of closed subsets of the given complementary sets. The homology and cohomology groups so obtained enter naturally into new exact homology and cohomology sequences.

BRST cohomology and invariants of four-dimensional gravity in Ashtekar variables.

We discuss the BRST cohomologies of the invariants associated with the description of classical and quantum gravity in four dimensions, using the Ashtekar variables. These invariants are constructed from several BRST cohomology sequences. They provide a systematic and clear characterization of non-local observables in general relativity with unbroken diffeomorphism invariance, and could yield further differential invariants for four-manifolds. The theory includes fluctuations of the vierbein fields, but there exits a non-trivial phase which can be expressed in terms of Witten's topological quantum field theory. In this phase, the descent sequences are degenerate, and the corresponding classical solutions can be identified with the conformally self-dual sector of Einstein manifolds. The full theory includes fluctuations which bring the system out of this sector while preserving diffeomorphism invariance.

Cohomology of the symplectic group 𝑆𝑝₄(𝑍). I. The odd torsion case

Let h 2 {h_2} be the degree two Siegel space and S p ( 4 , Z ) Sp(4,\mathbb {Z}) the symplectic group. The quotient S p ( 4 , Z ) ∖ h 2 Sp(4,\mathbb {Z})\backslash {h_2} can be interpreted as the moduli space of stable Riemann surfaces of genus 2 2 . This moduli space can be decomposed into two pieces corresponding to the moduli of degenerate and nondegenerate surfaces of genus 2 2 . The decomposition leads to a Mayer-Vietoris sequence in cohomology relating the cohomology of S p ( 4 , Z ) Sp(4,\mathbb {Z}) to the cohomology of the genus two mapping class group Γ 2 0 \Gamma _2^0 . Using this tool, the 3 3 - and 5 5 -primary pieces of the integral cohomology of S p ( 4 , Z ) Sp(4,\mathbb {Z}) are computed.

On the Cohomology Ring of Hyperplane Complements

Using only the long exact sequence of local cohomology, we give a brief derivation of Orlik and Solomon’s presentation for the cohomology ring of a complement of hyperplanes in a vector space.

On the cohomology ring of hyperplane complements

Using only the long exact sequence of local cohomology, we give a brief derivation of Orlik and Solomon’s presentation for the cohomology ring of a complement of hyperplanes in a vector space.

The adiabatic limit, Hodge cohomology and Leray's spectral sequence for a fibration

The adiabatic limit, Hodge cohomology and Leray's spectral sequence for a fibration

Whitney-sums (fibre-joins) in over space theory and obstruction theory for cohomology with local coefficients

SynopsisThe generalised Whitney sum (fibre-join) and the h-fibre-join can be defined in topM, the category of spaces over M. We note here some general properties of these constructions, and, as a specific example, we consider the relation between them and the extensions to the topM category of the top h-fibre-sequences F∗ΩB→E ∪ CF→B determined by top fibrations F→E→B. As an application we obtain the truncated local coefficient cohomology sequence for a top fibration which is topM principal fibration: this situation applies, for example, to the various stages of the Postnikov decomposition of a non-simply connected space X, and in this case we have M = K1(π1(X)).

The tower of n-groupoids and the long cohomology sequence

Given an exact category E , we associate to it a fibration c above E such that, for each object X of E , the fiber c[ x] is again exact. If, moreover, A is an internal abelian group in E , it determines a family of abelian groups A x in the fibres c[ x], such that the group H 1( E,A) is the colimit of the H 0( c[ x], A x ). This remark allows us to define iteratively H n+1( E,A) as the colimit of the H n ( c[ x], A x ). These groups are shown to have the property of the long cohomology sequence. When E=Ab , the construction coincides, up to isomorphism, with Yoneda's classical description of Ext n . When E=Grp , it coincides with the cohomology groups of a group in the sense of Eilenberg-Mac Lane.

Projective characters of degree one and the inflation-restriction sequence

AbstractLet G be a finite group, α be a fixed cocycle of G and Proj (G, α) denote the set of irreducible projective characters of G lying over the cocycle α.Suppose N is a normal subgroup of G. Then the author shows that there exists a G- invariant element of Proj(N, αN) of degree 1 if and only if [α] is an element of the image of the inflation homomorphism from M(G/N) into M(G), where M(G) denotes the Schur multiplier of G. However in many situations one can produce such G-invariant characters where it is not intrinsically obvious that the cocycle could be inflated. Because of this the author obtains a restatement of his original result using the Lyndon-Hochschild-Serre exact sequence of cohomology. This restatement not only resolves the apparent anomalies, but also yields as a corollary the well-known fact that the inflation-restriction sequence is exact when N is perfect.

Global anomalies and algebraic topology

The algebraic topology aspect of the global pure gauge anomaly calculation is investigated. In particular, the use of a cohomology sequence clarifies the method initiated by Witten [Nucl. Phys. 223, 422, 433 (1983)] and Elitzur and Nair [Nucl. Phys. B 243, 205 (1984)]. Examples in SU(N), Sp(N), and SO(N) are discussed.

Extensions of lattices overp-solvable groups

By PETER SCHMID Suppose G is a finite p-solvable group for some prime p. Let K be a finite extension of the p- adic number field 9 and k = R/r~R its residue class field. Let U, Vbe RG-lattices ( = R-free RG-modules of finite R-rank). The objective of this note is to record the following. Theorem. Ext~o (U, V) = 0 if either (i) K|174 is irreducible belonging to a block with cyclic defect group and K is unramified over •, or (ii) k| = k| is irreducible and the ramification index of K over 9 is less than p- 1 (and hence p odd). Statement (i) generalizes a result by Berman [11 who considered the case where G has a normal cyclic Sylow p-subgroup (and K = 9 It has been already noticed by Gudivok [31 that the result does not hold, in general, if K is ramified over 9 In a certain sense, both statements rely on the Fong-Swan theorem. In the first case we use the theory of blocks with cyclic defect groups and the fact that the Brauer tree is a star if G is p-solvable. The second statement follows from the uniqueness part of the lifting theorem in [61. Thus solvability is a crucial assumption. 1. Extensions of lattices. Let G be a finite group and let U, Vbe RG-lattices. Multiplication with 7r gives the exact sequence O-*V&V-,fz~O where f'= V/IrV~k| Since U is R- projective, the induced sequence 0--Hom R (U, V)&HomR (U, V)~Hom R (U, I7") ~0 is exact as well, and Extlo (U, V) = H 1 (G, Hom R (U, V)). Lemma. Ext~o(O,V)=O if and only if the residue class map HomRc(U, V)~ Homk6 (U, V) is epimorphic. P r o o f. Application of the long exact cohomology sequence yields the exact sequence 0 ~HomRc (U, V) ~ HomRo (U, V) --*Homko (0, ~2)-~Ext~o (U, V) Z*Ext~a (U, V) .... .

Almost short exact sequences in non-abelian cohomology

Almost short exact sequences in non-abelian cohomology

The second degree cohomology of finite orthogonal groups, II

Let V be a finite-dimensional vector space over GF(q), where q is odd. Suppose that dim V= 2m + 1, m = 1,2,..., and that V is equipped with a nondegenerate quadratic form Q. Let R(2m + 1, q)-or just Sa, when m and q are clear from the context--denote the commutator subgroup of the general orthogonal group of this quadratic form; that is, l2(2m + 1, q) is a Chevalley group of type B(m, q). In this paper we compute the second degree cohomology of n on its standard module V. In the smallest case (dim V= 3), H*(fi, V) is zero for q = 3, and it is of dimension 1 over GF(q), otherwise [lo]. Hence we will assume from now on that dim Va 5. Our main result is that, under these assumptions, H*(J2, V) is always zero, except for the group n(7,3), where we obtain an upper bound of 1 for the dimension of the second degree cohomology group over GF(3). The group H*(0(7, 3), V) is already known to be nonzero [5], so that it constitutes the only exception. The general outline of the proof is to calculate H*(G, V), where G is the maximal parabolic subgroup of R which is the stabilizer of a singular point. Since the restriction map from n to G induces an injection on the cohomology level, H*(R, V) will be trivial provided that H*(G, V) is. Except for the cases of q = 3, dim V= 5, 7 or 9, it is fairly straightforward to obtain the result that H’(G, V) is zero, by studying certain exact cohomology sequences obtained from the E,-terms of the Hochschild-Serre spectral sequence. In the two cases of q = 3, dim V= 5 or 9, where H*(G, V) is nonzero, we obtain explicit formulas for the nontrivial cocycle classes on the 3Sylow subgroup of the orthogonal group which is contained in the maximal parabolic subgroup G. We then show that these cocycle classes are not stable ones, hence in these two cases, H*(R, V) must be trivial again. 88 0021-8693/80/110088-22$02.00/O

Exact sequences in cohomology and an application

Exact sequences in cohomology and an application

The second degree cohomology of finite orthogonal groups, I

Let V be a finite-dimensional vector space over GF(q), where q is odd. Suppose that dim V= 2m + 1, m = 1,2,..., and that V is equipped with a nondegenerate quadratic form Q. Let R(2m + 1, q)-or just Sa, when m and q are clear from the context--denote the commutator subgroup of the general orthogonal group of this quadratic form; that is, l2(2m + 1, q) is a Chevalley group of type B(m, q). In this paper we compute the second degree cohomology of n on its standard module V. In the smallest case (dim V= 3), H*(fi, V) is zero for q = 3, and it is of dimension 1 over GF(q), otherwise [lo]. Hence we will assume from now on that dim Va 5. Our main result is that, under these assumptions, H*(J2, V) is always zero, except for the group n(7,3), where we obtain an upper bound of 1 for the dimension of the second degree cohomology group over GF(3). The group H*(0(7, 3), V) is already known to be nonzero [5], so that it constitutes the only exception. The general outline of the proof is to calculate H*(G, V), where G is the maximal parabolic subgroup of R which is the stabilizer of a singular point. Since the restriction map from n to G induces an injection on the cohomology level, H*(R, V) will be trivial provided that H*(G, V) is. Except for the cases of q = 3, dim V= 5, 7 or 9, it is fairly straightforward to obtain the result that H’(G, V) is zero, by studying certain exact cohomology sequences obtained from the E,-terms of the Hochschild-Serre spectral sequence. In the two cases of q = 3, dim V= 5 or 9, where H*(G, V) is nonzero, we obtain explicit formulas for the nontrivial cocycle classes on the 3Sylow subgroup of the orthogonal group which is contained in the maximal parabolic subgroup G. We then show that these cocycle classes are not stable ones, hence in these two cases, H*(R, V) must be trivial again. 88 0021-8693/80/110088-22$02.00/O

Quantum mechanics on topological networks

A quantum mechanical model of electrons on a network which has been used to obtain the line spectra of conjugated organic molecules and the band structure for various solid state systems is shown to be intimately connected with the homology and cohomology sequences of the algebraic topology of 1-complexes. Thus, the remarkable qualitative agreement of these models to experiment implies that those calculated physical quantities are structural quantities. Quantum mechanical conservation laws analogous to Kirchhoff’s electrical conservation laws are derived by variational methods applied to the system’s Lagrangian. The topology is shown to enforce energy conservation on the network.

The Spectral Sequence of a Finite Group Extension Stops

It is proved that if G is a finite group, H a normal subgroup, and A a finitely generated G-module, then both the cohomology and homology spectral sequences for the group extension stop in a finite number of stops. A lemma about ${\operatorname {Tor}}(M,N)$ as a module over $R \otimes S$ is proved. Two spectral sequences of Hochschild and Serre are shown to be the same.

The spectral sequence of a finite group extension stops

It is proved that if G is a finite group, H a normal subgroup, and A a finitely generated G-module, then both the cohomology and homology spectral sequences for the group extension stop in a finite number of stops. A lemma about Tor(M, N) as a module over R 0 S is proved. Two spectral sequences of Hochschild and Serre are shown to be