On the homology of elementary Abelian groups as modules over the Steenrod algebra

We examine the dual of the so-called “hit problem”, the latter being the problem of determining a minimal generating set for the cohomology of products of infinite projective spaces as a module over the Steenrod Algebra A at the prime 2. The dual problem is to determine the set of A-annihilated elements in homology. The set of A-annihilateds has been shown by David Anick to be a free associative algebra. In this note we prove that, for each k≥0, the set of kpartiallyA-annihilateds, the set of elements that are annihilated by Sqⁱ for each i≤2^k, itself forms a free associative algebra.     

The homotopy categorical crossed module of a CW-complex

Crossed modules have longstanding uses in homotopy theory and the cohomology of groups. The corresponding notion in the setting of categorical groups, that is, categorical crosses modules, allowed the development of a low-dimensional categorical group cohomology. Now, its relevance is also shown here to homotopy types by associating, to any pointed CW-complex (X,∗), a categorical crossed module that algebraically represents the homotopy 3-type of X.     

Module structures and the derived functors of iterated loop functors on unstable modules over the Steenrod algebra

The calculation of the iterated loop functors and their left derived functors on the category of unstable modules over the Steenrod algebra is a non-trivial problem; Singer constructed an explicit and functorial chain complex to calculate these functors. The results of Singer are analysed to give information on the behaviour of these functors with respect to the nilpotent filtration of the category of unstable modules.We show that, if an unstable module M supports an action of an unstable algebra K, then the derived functors of the iterated loop functors applied to M support actions of iterated doubles of K. This allows the finiteness results of Henn on unstable modules which support actions of unstable algebras to be applied to deduce structural results on the derived functors of iterated loops on such modules.     

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.     

Curved Koszul duality of algebras over unital versions of binary operads

We develop a curved Koszul duality theory for algebras presented by quadratic-linear-constant relations over unital versions of binary quadratic operads. As an application, we study Poisson n-algebras given by polynomial functions on a standard shifted symplectic space. We compute explicit resolutions of these algebras using curved Koszul duality. We use these resolutions to compute derived enveloping algebras and factorization homology on parallelized simply connected closed manifolds with coefficients in these Poisson n-algebras.     

Hall Numbers and the Composition Algebra of the Kronecker Algebra

We present formulas for the structure constants (Hall numbers) of the Hall algebra associated to the Kronecker algebra. The formulas which in some cases involve the classical Hall polynomials enable us to determine every Hall number. Using again these formulas we construct new PBW-bases with simple structure constants for the composition algebra , making possible the definition of the generic composition algebra via Hall polynomials.Presented by C. Ringel.     

Secondary derived functors and the Adams spectral sequence

Classical homological algebra takes place in additive categories. In homotopy theory such additive categories arise as homotopy categories of “additive groupoid enriched categories”, in which a secondary analog of homological algebra can be performed. We introduce secondary chain complexes and secondary resolutions leading to the concept of secondary derived functors. As a main result we show that the E₃-term of the Adams spectral sequence can be expressed as a secondary derived functor. This result can be used to compute the E₃-term explicitly by an algorithm.     

The bar derived category of a curved dg algebra

Curved A_∞-algebras appear in nature as deformations of dg algebras. We develop the basic theory of curved A_∞-algebras and, in particular, curved dg algebras. We investigate their link with a suitable class of dg coalgebras via the bar construction and produce Quillen model structures on their module categories. We define the analogue of the relative derived category for a curved dg algebra.     

The negative answer to Kameko's conjecture on the hit problem

The goal of this paper is to give a negative answer to Kameko's conjecture on the hit problem stating that the cardinal of a minimal set of generators for the polynomial algebra Pk, considered as a module over the Steenrod algebra A, is dominated by an explicit quantity depending on the number of the polynomial algebra's variables k. The conjecture was shown by Kameko himself for k?3 in his PhD thesis in the Johns Hopkins University in 1990, and recently proved by us and Kameko for k=4. However, we claim that it turns out to be wrong for any k>4.In order to deny Kameko's conjecture we study a minimal set of generators for A-module Pk in some so-call generic degrees. What we mean by generic degrees is a bit different from that of other authors in the fields such as Crabb-Hubbuck, Nam, Repka-Selick, Wood …We prove an inductive formula for the cardinal of the minimal set of generators in these generic degrees when the number of the variables, k, increases. As an immediate consequence of this inductive formula, we recognize that Kameko's conjecture is no longer true for any k>4.     

Topological André-Quillen Cohomology and E∞ André-Quillen Cohomology

We show that the André-Quillen cohomology of an E_∞ simplicial algebra with arbitrary coefficients and the topological André-Quillen cohomology of an E_∞ ring spectrum with Eilenberg-Mac Lane coefficients may be calculated as the André-Quillen cohomology of an associated E_∞ differential graded algebra.     

Bifurcation from periodic solutions with spatiotemporal symmetry, including resonances and mode interactions

We study local bifurcation in equivariant dynamical systems from periodic solutions with a mixture of spatial and spatiotemporal symmetries.In previous work, we focused primarily on codimension one bifurcations. In this paper, we show that the techniques used in the codimension one analysis can be extended to understand also higher codimension bifurcations, including resonant bifurcations and mode interactions. In particular, we present a general reduction scheme by which we relate bifurcations from periodic solutions to bifurcations from fixed points of twisted equivariant diffeomorphisms, which in turn are linked via normal form theory to bifurcations from equilibria of equivariant vector fields.We also obtain a general theory for bifurcation from relative periodic solutions and we show how to incorporate time-reversal symmetries into our framework.     

模p Steenrod代数上同调的一些注记

模p Steenrod代数A的上同调H~(s,t)(A)是决定球面稳定同伦群的最有力数据.首先给出了模p Steenrod代数A和May谱序列的一些重要结论,而后给出与乘积元γ_(s+3)l_ng_0∈H~(s+8,t(s,n))(A)密切相关的May谱E_1项的结果,这些结论对该乘积元的非平凡性研究有重要意义,其中t(s,n)=p~(n+1)q+2p~nq+(s+3)p~2q+(s+3)pq+(s+3)q+s, 0≤sp-6, n≥4, p≥11, q=2(p-1).     

Higher order derived functors and the Adams spectral sequence

Classical homological algebra studies chain complexes, resolutions, and derived functors in additive categories. In this paper we define higher order chain complexes, resolutions, and derived functors in the context of a new type of algebraic structure, called an algebra of left cubical balls  . We show that higher order resolutions exist in these algebras, and that they determine higher order Ext-groups. In particular, the _Em$E_m$-term of the Adams spectral sequence (m>2)$(m>2)$ is such a higher Ext-group, providing a new way of constructing its differentials.     

Near a local topological equivalence when a quasi-center like point appears

As cycles are very important in economic models, in this work we analyze what happens for a model given by a piecewise-linear discontinuous map defined on the plane. As the main result, it is proved this model is near to being (locally) topologically equivalent to a rotation. Besides, some counterexamples are provided to demonstrate that the hypotheses assumed are necessary.     

Cyclicity of a simple focus via the vanishing multiplicity of inverse integrating factors

First we provide new properties about the vanishing multiplicity of the inverse integrating factor of a planar analytic differential system at a focus. After we use this vanishing multiplicity for studying the cyclicity of foci with pure imaginary eigenvalues and with homogeneous nonlinearities of arbitrary degree having either its radial or angular speed independent of the angle variable in polar coordinates. After we study the cyclicity of a class of nilpotent foci in their analytic normal form.