Eilenberg-Moore Spectral Sequence Research Articles

Motivic cohomology and infinitesimal group schemes

For $k$ a perfect field of characteristic $p>0$ and $G/k$ a split reductive group with $p$ a non-torsion prime for $G,$ we compute the mod $p$ motivic cohomology of the geometric classifying space $BG_{(r)}$, where $G_{(r)}$ is the $r$th Frobenius kernel of $G.$ Our main tool is a motivic version of the Eilenberg-Moore spectral sequence, due to Krishna. For a flat affine group scheme $G/k$ of finite type, we define a cycle class map from the mod $p$ motivic cohomology of the classifying space $BG$ to the mod $p$ \'etale motivic cohomology of the classifying stack $\mathcal{B}G.$ This also gives a cycle class map into the Hodge cohomology of $\mathcal{B}G.$ We study the cycle class map for some examples, including Frobenius kernels.

Basic Cohomology of Canonical Holomorphic Foliations on Complex Moment-Angle Manifolds

Abstract We describe the basic cohomology ring of the canonical holomorphic foliation on a moment-angle manifold, LVMB-manifold, or any complex manifold with a maximal holomorphic torus action. Namely, we show that the basic cohomology has a description similar to the cohomology ring of a complete simplicial toric variety due to Danilov and Jurkiewicz. This settles a question of Battaglia and Zaffran, who previously computed the basic Betti numbers for the canonical holomorphic foliation in the case of a shellable fan. Our proof uses an Eilenberg–Moore spectral sequence argument; the key ingredient is the formality of the Cartan model for the torus action on a moment-angle manifold. We develop the concept of transverse equivalence as an important tool for studying smooth and holomorphic foliated manifolds. For an arbitrary complex manifold with a maximal torus action, we show that it is transverse equivalent to a moment-angle manifold and therefore has the same basic cohomology.

Simplicial cochain algebras for diffeological spaces

The original de Rham cohomology due to Souriau and the singular cohomology in diffeology are not isomorphic to each other in general. This manuscript introduces a singular de Rham complex endowed with an integration map into the singular cochain complex which gives the de Rham theorem for every diffeological space. It is also proved that a morphism called the factor map from the original de Rham complex to the new one is a quasi-isomorphism for a manifold and, more general, a space with singularities. Moreover, Chen’s iterated integrals are considered in a diffeological framework. As a consequence, we deduce that the bar complex of the original de Rham complex of a simply-connected diffeological space is quasi-isomorphic to the singular de Rham complex of the diffeological free loop space provided the factor map for the underlying diffeological space is a quasi-isomorphism. The process for proving the assertion yields the Leray–Serre spectral sequence and the Eilenberg–Moore spectral sequence in diffeology.

A new Kenzo module for computing the Eilenberg-Moore spectral sequence

In this work we present a new module for the computer algebraic topology system Kenzo computing the Eilenberg-Moore spectral sequence of fibrations between spaces with effective homology. These programs can be applied to determine the Eilenberg-Moore spectral sequence of extensions of finitely generated abelian groups.

Massey products, toric topology and combinatorics of polytopes

In this paper we introduce a direct family of simple polytopes such that for any there are non-trivial strictly defined Massey products of order in the cohomology rings of their moment-angle manifolds . We prove that the direct sequence of manifolds has the following properties: every manifold is a retract of , and one has inverse sequences in cohomology (over and , where as ) of the Massey products constructed. As an application we get that there are non-trivial differentials , for arbitrarily large as , in the Eilenberg–Moore spectral sequence connecting the rings and with coefficients in a field, where .

A new lower bound for LS-category

Let X be a simply connected CW-complex of finite type and {mathbb {K}} an arbitrary field. In this paper, we use the Eilenberg–Moore spectral sequence of C_*(Omega (X), mathbb K) to introduce a new homotopical invariant textsc {r}(X, {mathbb {K}}). If X is a Gorenstein space with nonzero evaluation map, then textsc {r}(X, {mathbb {K}}) turns out to interpolate mathrm {depth}(H_*(Omega (X), {mathbb {K}})) and mathrm {e}_{{mathbb {K}}}(X). We also define for any minimal Sullivan algebra (Lambda V,d) a new spectral sequence and make use of it to associate to any 1-connected commutative differential graded algebra (A, d) a similar invariant textsc {r}(A,d). When (Lambda V,d) is a minimal Sullivan model of X, this invariant fulfills the relation textsc {r}(X, {mathbb {K}}) = textsc {r}(Lambda V,d).

On non-realization results and conjectures of N. Kuhn

We discuss two extensions of results conjectured by Nick Kuhn about the non-realization of unstable algebras as the mod-p singular cohomology of a space, for p a prime. The first extends and refines earlier work of the second and fourth authors, using Lannes’ mapping space theorem. The second (for the prime 2) is based on an analysis of the -1 and -2 columns of the Eilenberg–Moore spectral sequence, and of the associated extension. In both cases, the statements and proofs use the relationship between the categories of unstable modules and functors between Fp-vector spaces. The second result in particular exhibits the power of the functorial approach.

Derived string topology and the Eilenberg-Moore spectral sequence

Let M be a simply-connected closed manifold of dimension m. Chas and Sullivan have defined (co)products on the homology of the free loop space H*(LM). Felix and Thomas have extended the loop (co)products to those of simply-connected Gorenstein spaces over a field. We describe these loop (co)products in terms of the torsion and extension functors by developing string topology in appropriate derived categories. In Algebraic Topology, one of the most important tools for computing the (co)homology of the space of free loops on a space is the (co)homological Eilenberg-Moore spectral sequence. Consider, over any field, the homological Eilenberg-Moore spectral sequence converging to H *(LM). Our description of the loop product enables one to conclude that this spectral sequence is multiplicative with respect to the Chas-Sullivan loop product and that its E 2-term is the Hochschild cohomology of H*(M). This gives a new method to compute the loop products on H *(LS m ) and H *(LℂP r ), the free loop space homology of spheres and complex projective spaces.

Fixed point adjunctions for equivariant module spectra

We consider the Quillen adjunction between fixed points and inflation in the context of equivariant module spectra over equivariant ring spectra, and give numerous examples including some based on geometric fixed points and some on the Eilenberg-Moore spectral sequence. These results were originally presented as part of our equivalence between rational torus-equivariant spectra and an algebraic model in arXiv:1101.2511. However, the present results apply in many other interesting cases explored here, which are not rational and where the ambient group is not a torus. The material in arXiv:1101.2511v3 will be revised to refer to this paper.

Behavior of the Eilenberg–Moore spectral sequence in derived string topology

The purpose of this paper is to give applications of the Eilenberg–Moore type spectral sequence converging to the relative loop homology algebra of a Gorenstein space, which is introduced in the previous paper due to the authors. Moreover, it is proved that the spectral sequence is functorial on the category of simply-connected Poincaré duality spaces over a space.

The Salvetti complex and the little cubes

For a real central arrangement $\mathcal{A}$, Salvetti introduced a construction of a finite complex Sal$(\mathcal{A})$ which is homotopy equivalent to the complement of the complexified arrangement in \[Sal87]. For the braid arrangement $\mathcal{A}{k-1}$, the Salvetti complex Sal$(\mathcal{A}{k-1})$ serves as a good combinatorial model for the homotopy type of the configuration space $F(\mathbb C,k)$ of $k$ points in $\mathbb C$, which is homotopy equivalent to the space $\mathcal{C}\_2(k)$ of $k$ little $2$-cubes. Motivated by the importance of little cubes in homotopy theory, especially in the study of iterated loop spaces, we study how the combinatorial structure of the Salvetti complexes of the braid arrangements are related to homotopy theoretic properties of iterated loop spaces. As a consequence, we prove the skeletal filtrations on the Salvetti complexes of the braid arrangements give rise to the cobar-type Eilenberg-Moore spectral sequence converging to the homology of $\Omega^2\Sigma^2 X$. We also construct a new spectral sequence that computes the homology of $\Omega^{\ell}\Sigma^{\ell} X$ for $\ell>2$ by using a higher order analogue of the Salvetti complex. The $E^1$-term of the spectral sequence is described in terms of the homology of $X$. The spectral sequence is different from known spectral sequences that compute the homology of iterated loop spaces, such as the Eilenberg-Moore spectral sequence and the spectral sequence studied by Ahearn and Kuhn in \[AK02].

Bockstein closed 2-group extensions and cohomology of quadratic maps

A central extension of the form E:0→V→G→W→0, where V and W are elementary abelian 2-groups, is called Bockstein closed if the components qi∈H⁎(W,F2) of the extension class of E generate an ideal which is closed under the Bockstein operator. In this paper, we study the cohomology ring of G when E is a Bockstein closed 2-power exact extension. The mod-2 cohomology ring of G has a simple form and it is easy to calculate. The main result of the paper is the calculation of the Bocksteins of the generators of the mod-2 cohomology ring using an Eilenberg–Moore spectral sequence. We also find an interpretation of the second page of the Bockstein spectral sequence in terms of a new cohomology theory that we define for Bockstein closed quadratic maps Q:W→V associated to the extensions E of the above form.

Galois theory and Lubin-Tate cochains on classifying spaces

We consider brave new cochain extensions F(BG +,R) → F(EG +,R), where R is either a Lubin-Tate spectrum E n or the related 2-periodic Morava K-theory K n , and G is a finite group. When R is an Eilenberg-Mac Lane spectrum, in some good cases such an extension is a G-Galois extension in the sense of John Rognes, but not always faithful. We prove that for E n and K n these extensions are always faithful in the K n local category. However, for a cyclic p-group \(C_{p^r } \), the cochain extension \(F(BC_{p^r + } ,E_n ) \to F(EC_{p^r + } ,E_n )\) is not a Galois extension because it ramifies. As a consequence, it follows that the E n -theory Eilenberg-Moore spectral sequence for G and BGdoes not always converge to its expected target.

Homology of iterated loop spaces of homogeneous spaces associated with exceptional Lie groups

We study the mod 2 homology of the double and triple loop spaces of homogeneous spaces associated with exceptional Lie groups. The main computational tools are the Serre spectral sequence for fibrations Ω n + 1 G → Ω n + 1 ( G / H ) → Ω n H for n = 1 , 2 , and the Eilenberg–Moore spectral sequence associated with related fiber squares, which both converge to the same destination space H ∗ ( Ω n ( G / H ) ; F 2 ) . We also develop the generalized Bockstein lemma to determine the higher Bockstein actions.

Cellular approximations and the Eilenberg–Moore spectral sequence

We set up machinery for recognizing k-cellular modules and k-cellular approximations, where k is an R-module and R is either a ring or a ring-spectrum. Using this machinery we can identify the target of the Eilenberg-Moore cohomology spectral sequence for a fibration in various cases. In this manner we get new proofs for known results concerning the Eilenberg-Moore spectral sequence and generalize another result.

Homology of gauge groups associated with special unitary groups

We study the homology of gauge groups associated with principal SU ( n ) bundles over the four-sphere. After computing the mod p homology of based gauge groups of SU ( n ) by combined use of the Serre spectral sequence and the Eilenberg–Moore spectral sequence, we compute the mod p homology of gauge groups of SU ( n ) using the Serre spectral sequence for the evaluation fibration.

HOMOLOGY OF BASED GAUGE GROUPS ASSOCIATED WITH SPINOR GROUP

We study the homology of based gauge groups associated with the principal Spin(n) bundles over the four-sphere using the Eilenberg-Moore spectral sequence and the Serre spectral sequence with the Dyer-Lash of operations.

Variations on the cohomology of loop spaces on generalized homogeneous spaces

The cohomology of the space of loops on generalized homogeneous spaces is determined by using the Eilenberg-Moore spectral sequence. This generalizes classical results for homogeneous spaces of compact Lie groups.

Weights in cohomology and the Eilenberg-Moore spectral sequence

We show that in the category of complex algebraic varieties, the Eilenberg--Moore spectral sequence can be endowed with a weight filtration. This implies that it degenerates if all involved spaces have pure cohomology. As application, we compute the rational cohomology of an algebraic $G$-variety $X$ ($G$ being a connected algebraic group) in terms of its equivariant cohomology provided that $H_G(X)$ is pure. This is the case, for example, if $X$ is smooth and has only finitely many orbits. We work in the category of mixed sheaves; therefore our results apply equally to (equivariant) intersection homology.

Eilenberg-Moore spectral sequence calculation of function space cohomology

The purpose of this article is to introduce an Eilenberg-Moore spectral sequence converging to the cohomology algebra of a function space with an adjunction space as its source. Computability of the spectral sequence is shown by determining explicitly the mod p cohomology algebra of the function space Open image in new window of maps from a non-orientable surface S to the classifying space of a simply-connected Lie group G whose homology is p-torsion free. Let M be a closed orientable 3-dimensional manifold. Applying the spectral sequence obtained from a Heegaard splitting of the manifold M, we also prove that Open image in new window is a direct summand of Open image in new window.