Let $p$ be an odd prime and ${\cal A}^{\ast}$ the mod $p$ Steenrod algebra. We study the technique known as "stripping'' applied to ${\cal A}^{\ast}$ and derive certain conjugation formulas both for ${\cal A}^{\ast}$ and its dual, generalising work of J. H. Silverman for $p=2$ ("Conjugation and excess in the Steenrod algebra", Proc. Am. Math. Soc. 119 (1993), no. 2, 657-661; "Hit polynomials and conjugation in the dual Steenrod algebra", Math. Proc. Camb. Philos. Soc. 123 (1998), no. 531-547) to the case of an odd prime.

We study the ring $\Gamma$ of all functions $\mathbb{N}^+ \to K$, endowed with the usual convolution product. $\Gamma$, which we call the ring of number-theoretic functions, is an adverse limit of the "truncations" $$\Gamma_n = \{ f \in \Gamma | \forall m > n : f(m) = 0 \}.$$ Each $\Gamma_n$ is a zero-dimensional, finitely generated $K$-algebra, which may be expressed as the quotient of a finitely generated polynomial ring with a stable (after reversing the order of the variables) monomial ideal. Using the description of the free minimal resolution of stable ideals given by Eliahou-Kervaire, and some additional arguments by Aramova-Herzog and Peeva, we give the Poincaré-Betti series for $\Gamma_n$.

We apply group cohomological methods to calculate the cohomology of $K(n)_*BP$ as a $K(n)_*K(n)$-comodule, recovering recent results of Hovey and Sadofsky. As applications we determine the Chromatic Spectral Sequence for $BP$ based on Johnson and Wilson's $E(n)$, showing the relationship to some generalizations of the classical Hattori-Stong Theorem and determine the change of Hopf algebroid spectral sequence associated with the natural map $BP\rightarrow E(n)$, extending calculations of Clarke for the Todd orientation $MU\rightarrow KU$.

This paper is prompted by work of Grandjeán and Ladra on homology crossed modules. We define the homology of a crossed module in dimensions one and two via the equivalence of categories with group objects in groupoids. We show that for any perfect crossed module, its second homology crossed module occurs as the kernel of its universal central extension as defined by Norrie. Our first homology is the same as that defined by Grandjeán and Ladra, but the second homology crossed modules are in general different. However, they coincide for aspherical perfect crossed modules. Our methods can in principle be applied to define a homology crossed module in any dimension.

In this paper, we prove various results concerning DGA-algebras in the context of the Homological Perturbation Theory. We distinguish two class of contractions for algebras: full algebra contractions and semi-full algebra contractions. A full algebra contraction is, in particular, a semi-full algebra contraction. Taking a full algebra contraction and an "algebra perturbation" as data of the Basic Perturbation Lemma, the Algebra Perturbation Lemma (or simply, F-APL) of [20] and [29 appears in a natural way. We establish here a perturbation machinery, the Semi-Full Algebra Perturbation Lemma (or, simply, SF-APL) that is a generalization of the previous one in the sense that the application range of SF-APL is wider than that of F-APL. We show four important applications in which this result is essential for the construction of algebra or coalgebra structures in various chain complexes.

In this paper, we define and investigate the properties of continuous family groupoids. This class of groupoids is necessary for investigating the groupoid index theory arising from the equivariant Atiyah-Singer index theorem for families, and is also required in noncommutative geometry. The class includes that of Lie groupoids, and the paper shows that, like Lie groupoids, continuous family groupoids always admit (an essentially unique) continuous left Haar system of smooth measures. We also show that the action of a continuous family groupoid $G$ on a continuous family $G$-space fibered over another continuous family $G$-space $Y$ can always be regarded as an action of the continuous family groupoid $G*Y$ on an ordinary $G*Y$-space.

Homology groups modulo $q$ of a precrossed $P$-module in any dimensions are defined in terms of nonabelian derived functors, where $q$ is a nonnegative integer. The Hopf formula is proved for the second homology group modulo $q$ of a precrossed $P$-module which shows that for $q=0$ our definition is a natural extension of Conduché and Ellis' definition [CE]. Some other properties of homologies of precrossed $P$-modules are investigated.

Recent results of Voevodsky and others have effectively led to the proof of the Lichtenbaum-Quillen conjectures at the prime 2, and consequently made it possible to determine the 2-homotopy type of the $K$-theory spectra for various number rings. The basic case is that of $ BGL({\Bbb Z})$; in this note we use these results to determine the 2-local (topological) $K$-theory of the space $BGL({\Bbb Z})$, which can be described as a completed tensor product of two quite simple components; one corresponds to a real `image of $J$' space, the other to $BBSO$.

Let complex $C^*$ algebras be endowed with a norm-continuous action of a fixed compact second countable group. From a separable $C^*$-algebra $A$ and a $\sigma $-unital $C^{*}$-algebra $B$, we construct a $C^{*}$-category Rep ($A,B$) and an isomorphism \[\kappa :K^{i+1}(\text{Rep} (A,B))\rightarrow KK^i(A,B),\;\;\;i\in \mathbb{Z}_2,\] where on the left-hand side are Karoubi's topological $K$-groups, and on the right-hand side are Kasparov's equivariant bivariant $K$-groups.

To any additive category $\mathfrak{C}$, we associate in a functorial way two additive categories $\mathcal {A}\mathfrak{C})$, $\mathcal B(\mathfrak{C})$. The category $\mathcal {A}(\mathfrak{C})$, resp. $\mathcal {B}(\mathfrak{C})$, is the reflection of $\mathfrak{C}$ in the category of additive categories with cokernels, resp. kernels, and cokernel, resp. kernel, preserving functors. Then the iteration $\mathcal {A}\mathcal {B}(\mathfrak{C})$ is the reflection of $\mathfrak{C}$ in the category of abelian categories and exact functors. We call $\mathcal {A}(\mathfrak{C})$ and $\mathcal {B}\mathfrak{C})$ the Freyd categories of $\mathfrak{C}$ since the first systematic study of these categories was done by Freyd in the mid-sixties. The purpose of the paper is to study further the Freyd categories and to indicate their applications to the module theory of an abelian or triangulated category.