The coarse shape groups

The (pointed) coarse shape category Sh^* (), having (pointed) topological spaces as objects and having the (pointed) shape category as a subcategory, was recently constructed. Its isomorphisms classify (pointed) topological spaces strictly coarser than the (pointed) shape type classification. In this paper we introduce a new algebraic coarse shape invariant which is an invariant of shape and homotopy, as well. For every pointed space (X,?) and for every k∈N₀, the coarse shape group , having the standard shape group for its subgroup, is defined. Furthermore, a functor is constructed. The coarse shape and shape groups already differ on the class of polyhedra. An explicit formula for computing coarse shape groups of polyhedra is given. The coarse shape groups give us more information than the shape groups. Generally, does not imply (e.g. for solenoids), but from pro-πk(X,?)=0 follows . Moreover, for pointed metric compacta (X,?), the n-shape connectedness is characterized by , for every k?n.     

Cohomology rings of multiplicative fibrations

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 imf^* and the structure of subkerf^* as modules over the Steenrod algebra.     

Cap pairings and spectral sequences

Let be a small category. For an -diagram X and -diagrams A and B of pointed spaces, each pairing X∧A→B satisfying the projection formula induces a pairing . In this note we show that there is an induced pairing of homotopy spectral sequences compatible with abutments in the sense that

     

Well-posedness of a dissipative nonlinear electrohydrodynamic system in modulation spaces

In this paper we prove the local and global well-posedness of a dissipative nonlinear electrohydrodynamic system in modulation spaces under certain conditions of s, q, and σ.     

Evaluation maps in rational homotopy

In the rational category of nilpotent complexes, let E be an H-space acting on a space X. With mild hypotheses we show that the action on the base point factors through a map Γ_E:S_E→X, where S_E is a finite product of odd-dimensional spheres and Γ_E is a homotopy monomorphism. Among others, the following consequences are obtained: if and only if is essential and if and only if X satisfies a strong splitting condition.     

Homotopy monomorphisms and H-splitting in loop space fibrations

Let be a fibration, the holonomy action of this fibration and the connecting map. It is shown that if the fibre F admits an H-structure ν such that ρ?ν○(1×∂) (principal fibrations of all kinds satisfy such a condition), then i is a monomorphism if and only if it is weak monomorphism, the latter is equivalent to that Ωp has a homotopy right inverse Γ. If in addition Γ is an H-map, then ΩE has the same H-type as ΩB×ΩF.     

Equivariant homotopical homology with coefficients in a Mackey functor

Let M be a Mackey functor for a finite group G. In this paper, generalizing the Dold-Thom construction, we construct an ordinary equivariant homotopical homology theory with coefficients in M, whose values on the category of finite G-sets realize the bifunctor M, both covariantly and contravariantly. Furthermore, we extend the contravariant functor to define a transfer in the theory for G-equivariant covering maps. This transfer is given by a continuous homomorphism between topological abelian groups.We prove a formula for the composite of the transfer and the projection of a G-equivariant covering map and characterize those Mackey functors M for which that formula has an expression analogous to the classical one.     

Torsion primes in loop space homology

We construct a finite 1-connected CW complex X such that has p-torsion for the infinitely many primes satisfying , but no p-torsion for the infinitely many primes satisfying p≡13 or .     

Blow-up of solutions to a degenerate parabolic equation not in divergence form

We study nonglobal positive solutions to the Dirichlet problem for u_t=u^p(Δu+u) in bounded domains, where 0<p<2. It is proved that the set of points at which u blows up has positive measure and the blow-up rate is exactly . If either the space dimension is one or p<1, the ω-limit set of consists of continuous functions solving . In one space dimension it is shown that actually as t→T, where w coincides with an element of a one-parameter family of functions inside each component of its positivity set; furthermore, we study the size of the components of {w>0} with the result that this size is uniquely determined by Ω in the case p<1, while for p>1, the positivity set can have the maximum possible size for certain initial data, but it may also be arbitrarily close to the minimal length π.     

On primitive roots for Carlitz modules

Let be the polynomial ring over a finite field. We prove that for every element a of a global -field of finite -characteristic the set of places for which a is a primitive root under the Carlitz action possesses a Dirichlet density. We also give a criterion for this density to be positive. This is an analogue of Bilharz’ version of the primitive roots conjecture of Artin, with replaced by the Carlitz module.     

Iterated homotopy fixed points for the Lubin-Tate spectrum

When G is a profinite group and H and K are closed subgroups, with H normal in K, it is not known, in general, how to form the iterated homotopy fixed point spectrum (ZhH)^hK/H, where Z is a continuous G-spectrum and all group actions are to be continuous. However, we show that, if G=Gn, the extended Morava stabilizer group, and , where is Bousfield localization with respect to Morava K-theory, En is the Lubin-Tate spectrum, and X is any spectrum with trivial Gn-action, then the iterated homotopy fixed point spectrum can always be constructed. Also, we show that is just , extending a result of Devinatz and Hopkins.     

On the product system of a completely positive semigroup

Given a W∗-continuous semigroup φ of unital, normal, completely positive maps of B(H), we introduce its continuous tensor product system . If α is a minimal dilation E₀-semigroup of φ with Arveson product system F, then is canonically isomorphic to F. We apply this construction to a class of semigroups of arising from a modified Weyl-Moyal quantization of convolution semigroups of Borel probability measures on . This class includes the heat flow on the CCR algebra studied recently by Arveson. We prove that the minimal dilations of all such semigroups are completely spatial, and additionally, we prove that the minimal dilation of the heat flow is cocyle conjugate to the CAR/CCR flow of index two.     

Characteristic classes and transversality

Let ξ be a smooth vector bundle over a differentiable manifold M. Let be a generic bundle morphism from the trivial bundle of rank n−i+1 to ξ. We give a geometric construction of the Stiefel-Whitney classes when ξ is a real vector bundle, and of the Chern classes when ξ is a complex vector bundle. Using h we define a differentiable closed manifold and a map whose image is the singular set of h. The ith characteristic class of ξ is the Poincaré dual of the image, under the homomorphism induced in homology by ?, of the fundamental class of the manifold . We extend this definition for vector bundles over a paracompact space, using that the universal bundle is filtered by smooth vector bundles.