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A well known quotient of the real Stiefel manifold is the projective Stiefel manifold. We introduce a new family of quotients of the real Stiefel manifold by cyclic group of order 2 whose action is induced by simultaneous pairwise flipping of the coordinates. We obtain a description for their tangent bundles, compute their mod 2 cohomology and compute the Stiefel Whitney classes of these manifolds. We use these to provide applications to their stable span, parallelizability and equivariant maps, and some associated results in topological combinatorics. [ABSTRACT FROM AUTHOR]

In this paper, we construct and study a Serre-type spectral sequence for motivic cohomology associated with a map of bisimplicial schemes with motivically cellular fiber. Then, we show how to apply it in order to approach the computation of the motivic cohomology of the Nisnevich classifying space of projective general linear groups. This naturally yields an explicit description of the motive of a Severi-Brauer variety in terms of twisted motives of its Cech simplicial scheme. [ABSTRACT FROM AUTHOR]

Dold manifolds $P(m,n)$ are certain twisted complex projective space bundles over real projective spaces and serve as generators for the unoriented cobordism algebra of smooth manifolds. The paper investigates the structure of finite groups that act freely on products of Dold manifolds. It is proved that if a finite group G acts freely and $ \mathbb{Z}_2 $ cohomologically trivially on a finite CW-complex homotopy equivalent to ${\prod_{i=1}^{k} P(2m_i,n_i)}$ , then $G\cong (\mathbb{Z}_2)^l$ for some $l\leq k$ (see Theorem A for the exact bound). We also determine some bounds in the case when for each i , n i is even and m i is arbitrary. As a consequence, the free rank of symmetry of these manifolds is determined for cohomologically trivial actions. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring where 2 is invertible. We compute the R -cohomology ring of the configuration space Conf ( R P m , k ) of k ordered points in the m -dimensional real projective space R P m . The method is based on the fact that the orbit configuration space of k ordered points in the m -dimensional sphere (with respect to the antipodal action) is a 2 k -fold covering of Conf ( R P m , k ) . This implies that, for odd m , the Leray spectral sequence for the inclusion Conf ( R P m , k ) ⊂ ( R P m ) k collapses after its first non-trivial differential, just as it does when R P m is replaced by a complex projective variety. The method also allows us to handle the R -cohomology ring of the configuration space of k ordered points in the punctured manifold R P m − ⋆ . [ABSTRACT FROM AUTHOR]

Abstract: In this note we define fibrations of topological stacks and establish their main properties. When restricted to topological spaces, our notion of fibration coincides with the classical one. We prove various standard results about fibrations (long exact sequence for homotopy groups, Leray–Serre and Eilenberg–Moore spectral sequences, etc.). We prove various criteria for a morphism of topological stacks to be a fibration, and use these to produce examples of fibrations. We prove that every morphism of topological stacks factors through a fibration and construct the homotopy fiber of a morphism of topological stacks. As an immediate consequence of the machinery we develop, we also prove van Kampenʼs theorem for fundamental groups of topological stacks. [Copyright &y& Elsevier]

We determine the mod p cohomology of the loop space and the double loop space of the symmetric space of exceptional type EI exploiting the Serre spectral sequence and the Eilenberg--Moore spectral sequence. [ABSTRACT FROM AUTHOR]

Abstract: Let denote an integral representation of the cyclic group of prime order p. This induces a -action on the torus . The goal of this paper is to explicitly compute the cohomology groups for any such representation. As a consequence we obtain an explicit calculation of the integral cohomology of the classifying space associated to the family of finite subgroups for any crystallographic group with prime holonomy. [Copyright &y& Elsevier]

Abstract: 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 in which a subspace, a discriminant, often an arrangement of linear subspaces , 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 , we can pose the problem of the existence of an equivariant mapping from X to the complement of the discriminant in . 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 , [•] 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. [Copyright &y& Elsevier]

Abstract: We compute the complete Fadell–Husseini index of the dihedral group acting on for and for coefficients, that is, the kernels of the maps in equivariant cohomology and This establishes the complete cohomological lower bounds, with and with coefficients, for the two-hyperplane case of Grünbaumʼs 1960 mass partition problem: For which d and j can any j arbitrary measures be cut into four equal parts each by two suitably chosen hyperplanes in ? In both cases, we find that the ideal bounds are not stronger than previously established bounds based on one of the maximal abelian subgroups of . [Copyright &y& Elsevier]

Abstract: We prove that any continuous map of an N-dimensional simplex with colored vertices to a d-dimensional manifold M must map r points from disjoint rainbow faces of to the same point in M: For this we have to assume that , no r vertices of get the same color, and our proof needs that r is a prime. A face of is a rainbow face if all vertices have different colors. This result is an extension of our recent “new colored Tverberg theorem”, the special case of . It is also a generalization of Volovikovʼs 1996 topological Tverberg theorem for maps to manifolds, which arises when all color classes have size 1 (i.e., without color constraints); for this special case Volovikovʼs proof, as well as ours, works when r is a prime power. [Copyright &y& Elsevier]

Abstract: Let p be an odd prime. Let be the fibre space induced from an H-map 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 as a ring. The ring depends on the structure of and the structure of as modules over the Steenrod algebra. [Copyright &y& Elsevier]

Let G be a compact Lie group. In 1960, P A Smith asked the following question: "Is it true that for any smooth action of G on a homotopy sphere with exactly two fixed points, the tangent G-modules at these two points are isomorphic?" A result due to Atiyah and Bott proves that the answer is 'yes' for ℤp and it is also known to be the same for connected Lie groups. In this work, we prove that two linear torus actions on Sn which are c-cobordant (cobordism in which inclusion of each boundary component induces isomorphisms in ℤ-cohomology) must be linearly equivalent. As a corollary, for connected case, we prove a variant of Smith's question. [ABSTRACT FROM AUTHOR]

Abstract: Let 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. [Copyright &y& Elsevier]

Abstract: We study the homology of gauge groups associated with principal bundles over the four-sphere. After computing the mod p homology of based gauge groups of by combined use of the Serre spectral sequence and the Eilenberg–Moore spectral sequence, we compute the modp homology of gauge groups of using the Serre spectral sequence for the evaluation fibration. [Copyright &y& Elsevier]

We study the homology groups of semicubical sets with coefficients in the homological systems of abelian groups. The main theorem states that the groups under consideration are isomorphic to the homology groups of the category of singular cubes. This yields an isomorphism criterion for the homology groups of semicubical sets, the spectral sequence of a locally directed covering, and the spectral sequence of a morphism of semicubical sets. [ABSTRACT FROM AUTHOR]

Abstract: John McCleary insisted in his interesting textbook entitled “User’s guide to spectral sequences” on the fact that the tool “spectral sequence” is not in the general situation an algorithm allowing its user to compute the looked-for homology groups. The present article explains how the notion of “Object with Effective Homology” on the contrary allows the user to recursively obtain all the components of the Serre and Eilenberg–Moore spectral sequences, when the data are objects with effective homology. In particular the computability problem of the higher differentials is solved, the extension problem at abutment is also recursively solved. Furthermore, these methods have been concretely implemented as an extension of the Kenzo computer program. Two typical examples of spectral sequence computations are reported. [Copyright &y& Elsevier]