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We study the mod p homology of the gauge group G k ( F 4 ) associated with a principal F 4 bundle by computing the Serre spectral sequence for the fibration Ω 0 4 ( F 4 ) → G k ( F 4 ) → F 4 . The main result is that the Serre spectral sequence collapses at the E 2 -term except for p = 2 , 13 .

The purpose of this article is to compute the mod 2 cohomology of Γ q ( K ) , the mapping class group of the Klein bottle with q marked points. We provide a concrete construction of Eilenberg–MacLane spaces X q = K ( Γ q ( K ) , 1 ) and fiber bundles F q ( K ) / Σ q → X q → B ( Z 2 × O ( 2 ) ) , where F q ( K ) / Σ q denotes the configuration space of unordered q-tuples of distinct points in K and B ( Z 2 × O ( 2 ) ) is the classifying space of the group Z 2 × O ( 2 ) . Moreover, we show the mod 2 Serre spectral sequence of the bundle above collapses.

The purpose of this article is to compute the cohomology of B Diff ( K ) , the classifying space of the diffeomorphisms of the Klein bottle. We analyze the Serre spectral sequence of the fibration B Diff 0 ( K ) → B Diff ( K ) → B ( Z 2 × Z 2 ) , with non-trivial coefficients, and show it collapses. Moreover, we show that B Diff ( K ) has the homotopy type of B Z 2 × B O ( 2 ) . Finally we give direct and concrete constructions for the Eilenberg–MacLane spaces K ( Γ q ( K ) , 1 ) , where Γ q ( K ) denotes the mapping class group of K with marked points π 0 Diff ( K ; q ) .

We use duality in the manner of Serre to generalize a theorem of Hedenmalm on solution of the $\bar \partial $ equation with inverse of the weight in Hormander $\displaystyle L^{2}$ estimates.\

The principal aim of this paper is to extend Abel's theorem to the setting of complex supermanifolds of dimension 1|q over a finite-dimensional local supercommutative C-algebra. The theorem is proved by establishing a compatibility of Serre duality for the supercurve with Poincare duality on the reduced curve. We include an elementary algebraic proof of the requisite form of Serre duality, closely based on the account of the reduced case given by Serre in Algebraic Groups and Class Fields, combined with an invariance result for the topology on the dual of the space of repartitions. Our Abel map, taking Cartier divisors of degree zero to the dual of the space of sections of the Berezinian sheaf, modulo periods, is defined via Penkov's characterization of the Berezinian sheaf as the cohomology of the de Rham complex of the sheaf D of differential operators, as a right module over itself. We discuss the Jacobi inversion problem for the Abel map and give an example demonstrating that if n is an integer sufficiently large that the generic divisor of degree n is linearly equivalent to an effective divisor, this need not be the case for all divisors of degree n., 14 pages

We study the behaviour of fibre maps under exponentiation, i.e. given a fibration p : E → B we ask for which spaces X is the induced map between mapping spaces p ⁎ : E X → B X also a fibration. If X is a locally compact space, the positive answer follows easily by the exponential law so in this paper we consider more general spaces and show that the preservation of fibrations is related to the local homotopy properties of the space X. For example, if p is a Dold fibration and X admits a deformation retraction on a compactly generated space, then the induced map p ⁎ is also a Dold fibration. Similar results hold for Hurewicz fibrations with unique path-lifting and for covering spaces, and can be furthermore extended to spaces that admit some numerable cover, whose elements preserve fibration property.

Algebraic topological methods are especially well suited for determining the non-existence of continuous mappings satisfying certain properties. In combinatorial problems it is sometimes possible to define a mapping from a space X of configurations to a Euclidean space R m in which a subspace, a discriminant, often an arrangement of linear subspaces A , expresses a target condition on the configurations. Add symmetries of all these data under a group G for which the mapping is equivariant. If we remove the discriminant from R m , we can pose the problem of the existence of an equivariant mapping from X to the complement of the discriminant in R m . Algebraic topology may sometimes be applied to show that no such mapping exists, and hence the image of the original equivariant mapping must meet the discriminant. We introduce a general framework, based on a comparison of Leray–Serre spectral sequences. This comparison can be related to the theory of the Fadell–Husseini index. We apply the framework to: • solve a mass partition problem ( antipodal cheeses ) in R d , • determine the existence of a class of inscribed 5-element sets on a deformed 2-sphere, • obtain two different generalizations of the theorem of Dold for the non-existence of equivariant maps which generalizes the Borsuk–Ulam theorem.

Let p be an odd prime. Let X˜ be the fibre space induced from an H-map f:X→K where K is a generalized Eilenberg MacLane space and X is a simply connected H-space. Such spaces occur frequently in Postnikov towers and connective covers. In this paper, we compute the mod p cohomology of X˜ as a ring. The ring depends on the structure of imf* and the structure of subkerf* as modules over the Steenrod algebra.

Let γ : S 1 → R 2 be a Jordan curve in the plane. It is a simple topological riddle to determine if there is an equilateral triangle with vertices on γ. By reformulating this question in the paradigm of configuration spaces and test maps, we can solve this riddle using a Borsuk–Ulam type theorem obtained using equivariant methods.

The membership of the local cohomology modules Han(M) of a module M in certain Serre subcategories of the category of modules is studied from below (in). Generalizations of depth and regular sequences are defined. The relation of these notions to local cohomology are found. It is shown that the membership of the local cohomology modules of a finite module in a Serre subcategory in the upper range just depends on the support of the module.

It was proved by H. Whitney in 1933 that a Serre fibration of compact metric spaces admits a global section provided every fiber is homeomorphic to the unit interval [ 0 , 1 ] . Results of this paper extend Whitney's theorem to the case when all fibers are homeomorphic to a given compact two-dimensional manifold.

Given a positively graded commutative coherent ring A which is finitely generated as an A_0-algebra, a bijection between the tensor Serre subcategories of qgr A and the set of all subsets Y\subseteq Proj A of the form Y=\bigcup_{i\in\Omega}Y_i with quasi-compact open complement Proj A\Y_i for all i\in\Omega is established. To construct this correspondence, properties of the Ziegler and Zariski topologies on the set of isomorphism classes of indecomposable injective graded modules are used in an essential way. Also, there is constructed an isomorphism of ringed spaces (Proj A,O_{Proj A}) --> (Spec(qgr A),O_{qgr A}), where (Spec(qgr A),O_{qgr A}) is a ringed space associated to the lattice L_{serre}(qgr A) of tensor Serre subcategories of qgr A., Comment: some minor corrections made

The purpose of this paper is to prove that the shriek map associated to a finite codimensional sub-fiberwise embedding between Hilbert manifolds behaves properly in regard of the associated Serre Spectral sequences. We apply this result to evaluate the Chas–Sullivan loop product of the total space of a fibration. Then, we compute up to extension issues the loop homology of sphere bundle of spheres.

We describe in detail Serre's application of spectral sequence theory to the study of the relations between the homology of total space, base space and fibre in a Serre fibration; and we apply the results to establish that a 1-connected space X has homology groups (in positive dimension) in a Serre class C if and only if its homotopy groups are in C . We include in this paper some personal reflections on the contact the author had with Serre during the decade of the 1950's when Serre's revolutionary work in homotopy theory was completely changing the face of algebraic topology.

E.H. Spanier (1992) has constructed, for a cohomology theory defined on a triangulated space and locally constant on each open simplex, a spectral sequence whose E 2 -term consists of certain simplicial cohomology groups, converging to the cohomology of the space. In this paper we study a closed G -fibration ƒ: Y → X , where G is a finite group. We show that if the base- G -space X is equivariantly triangulated and Y is paracompact, then Spanier's spectral sequence yields an equivariant Serre spectral sequence for ƒ. The main point here is to identify the equivariant singular cohomology groups of X with appropriate simplicial cohomology groups of the orbit space X G .

Given a finite degree field extension E ∣ F of nonfinite fields we show homologically BGl ( E ) can be viewed as the total space of fibration with fibre BGl ( F ). Associated with this fibration is a spectral sequence relating the homology of stable linear groups over field extensions.

Motivation One of the most convenient computational tools for the mod p homology of loop spaces is the cobar-type Eilenberg-Moore spectral sequence (EMSS, for short). The computation of H,(P’C”X; F,) using the path loop fibration nnznx -+ pn”-‘x:“x + R”_‘Z”_X (1) illustrates the power of EMSS. In this case, the spectral sequence E2 = CotorH*(R’~‘r”x;F,)(Fp, F,) =sH,(QTX; F,) collapses at E2, reducing the calculation to an easy homological algebra. On the other hand, the Serre spectral sequence for the fibration (1) contains infinitely many nontrivial differentials and does not compute H,(Q”C”X; F,) directly. It would be very useful to have an EMSS for any generalized homology theory, for example, Morava K-theory.

Let k be any field and let k-0-dga(*) be the category of connected cochain k-algebras. We define the notion of a twisted extension in this category, which generalizes the classical notion of a KS-extension. A morphism in k-0-dga(*) is called an algebraic fibration if it has the weak homotopy type of a twisted extension. An algebraic fibration may possess ''fibers'' with different cohomology algebras, though all with the same cohomology as vector space. Our main result is that, for any fibration F --> E --> B between 1-connected topological spaces of finite type, the induced map on the (reduced) cochain algebras C-*(B; k) --> C-*(E; k) is an algebraic fibration, and that one of the algebraic fibers has the weak homotopy type of C-*(F; k).

In this paper McGibbon-Neisendorfer's Serre Conjecture Theorem is improved. As corollaries, we apply our results to the homotopy groups of H-spaces.

We consider the Lyndon-Hochschild-Serre spectral sequence with Fn coefficients for a central extension with kernel cyclic of order a power of p and arbitrary discrete quotient group. For this spectral sequence d2 and d3 are known, and we give a description for d4. Using this result we deduce a similar formula for the Serre spectral sequence for a fibration with fibre BCpn. The differential from odd rows to even rows involves a Massey triple product, so we describe the calculation of such products in the cohomology of a finite abelian group. As an example we determine the Poincaré series for the mod-3 cohomology of various 3-groups.

We compute the filling radius with rational coefficients of C P n by studying the Serre spectral sequence of the total space of the unit tangent bundle viewed as a principal SO(3)-bundle on the Grassmannian of 2-planes in C P n . We also compute the (integer) filling radius of C P n and exhibit a torsion obstruction to filling C P n . We discuss a related aspect of the global geometry of C P n .

Let Λ ∗ be a not necessarily connected differential graded algebra. To each Λ ∗ -differential graded module we associate ‘characteristic’ classes which are invariants of the quasi-isomorphism class of this module and determine the Pontrjagin product by the 0th and the first homology group of Λ ∗ . We use them to determine the differential d 2 of homology spectral sequences such as the Serre spectral sequence or the Borel equivariant spectral sequence. This is done without classical hypothesis such as a one-connected base or a connected group.

We consider a quasi-compact ringed space (X,) for which the sheaf is flasque and locally Serre. We prove that for such a space the functor of global sections Γ defines an equivalence of the category of locally free left -modules of finite rank and the category of projective finitely generated Γ()-modules. Then we give a condition on an open covering U of X sufficient for every -module which is free on U to be free. If U satisfies this condition and besides is flaque and Serre on U then it follows that Γ() is also Serre. As an application we obtain a proof of Vorst's theorem that the face ring of an arbitrary finite complex is Serre.