Let $k$ be a field and $I$ an $H$-unital two-sided ideal of an associative unital $k$-algebra $A$. Then, based on Wodzicki's work, we show that the Hochschild homology groups of $A$ with coefficients in $I$ are isomorphic to the Hochschild homology groups of $I$.

In this note we apply the Guillén-Navarro descent theorem to define a descent variant of the algebraic $K$-theory of varieties over a field of characteristic zero, $KD(X)$, which coincides with $K(X)$ for smooth varieties and to prove that there is a natural weight filtration on the groups $KD*(X)$. After a result of Haesemeyer, we deduce that this theory is equivalent to the homotopy algebraic $K$-theory introduced by Weibel.

We study Ext-groups in the category of strict polynomial functors. The main result is a computation of the Ext-groups between twisted Weyl and Schur functors for a large class of Young diagrams.

We study complexes of finite complete intersection dimension in the derived category of a local ring. Given such a complex of complexity $c$, we prove that the thick subcategory it generates contains complexes of all possible complexities at most $c$. In particular, we show that such a complex is virtually small, answering a question raised by Dwyer, Greenlees and Iyengar.

We construct a combinatorial model of an $A∞$-operad which acts simplicially on the cobar resolution (not just its total space) of a simplicial set with respect to a ring $R$.

We provide a new algebraic description of the structure on the set of all unstable cohomology operations for a suitable generalised cohomology theory, $E*(–)$. Our description is as a graded and completed version of a Tall-Wraith monoid. The $E*$-cohomology of a space $X$ is a module for this Tall-Wraith monoid. We also show that the corresponding Hopf ring of unstable co-operations is a module for the Tall-Wraith monoid of unstable operations. Further examples are provided by considering operations from one theory to another.

Using the ku- and BP-theoretic versions of Astey's cobordism obstruction for the existence of smooth Euclidean embeddings of stably almost complex manifolds, we prove that, for e greater than or equal to α(n), the (2n+1)-dimensional 2e-torsion lens space cannot be embedded in Euclidean space of dimension 4n-2α(n)+1. (Here α(n) denotes the number of ones in the dyadic expansion of a positive integer n.) A slightly restricted version of this fact holds for e < α(n). We also give an inductive construction of Euclidean embeddings for 2e-torsion lens spaces. Some of our best embeddings are within one dimension of being optimal.

We introduce the notion of a 3-crossed module, which extends the notions of a 1-crossed module (Whitehead) and a 2-crossed module (Conduché). We show that the category of 3-crossed modules is equivalent to the category of simplicial groups having a Moore complex of length 3. We make explicit the relationship with the cat3-groups (Loday) and the 3-hypercomplexes (Cegarra-Carrasco), which also model algebraically homotopy 4-types.

We show that the free E∞-algebra on a zero-cell cannot be modeled by a commutative Γ-ring. The proof shows that Dyer-Lashof operations of positive degree must vanish on the zero'th homology of such an object.

In this paper we systematically describe relations between various structure sets which arise naturally for pairs of compact topological manifolds with boundary. Our consideration is based on a deep analogy between the case of a compact manifold with boundary and the case of a closed manifold pair. This approach also gives a possibility to construct the obstruction groups for natural maps of various structure sets and to investigate their properties.

Extending a result of McCord, we prove that every finite simplicial complex is homotopy equivalent to the inverse limit of a sequence of finite spaces. In addition to generalizing McCord's theorem, this provides it with a more geometric motivation, demonstrating a sense in which the simplicial complex is successively better approximated by its finite models.

This paper expands on and refines some known and less well-known results about the finite subset spaces of a simplicial complex X including their connectivity and manifold structure. It also discusses the inclusion of the singletons into the three-fold subset space and shows that this subspace is weakly contractible but generally non-contractible unless X is a cogroup. Some homological calculations are provided.

Working over an algebraically closed field of characteristic zero, we compute the cohomology of the subalgebra A(2) of the motivic Steenrod algebra that is generated by Sq1, Sq2, and Sq4. The method of calculation is a motivic version of the May spectral sequence.

Speculatively assuming that there is a "motivic modular forms" spectrum with certain properties, we use an Adams-Novikov spectral sequence to compute the homotopy of such a spectrum at the prime 2.

Let X, Y be finite 1-connected p-local CW-complexes. We show that cat(X×Y)=cat(X)+cat(Y) holds if p is large and the loop space homology of X, Y satisfies a certain flatness assumption.

The aim of this paper is to prove that the category of cubes with connections, introduced by R. Brown and Ph.J. Higgins, is a strict test category in Grothendieck's sense. In particular this implies that cubical sets with connections are models for homotopy types in a very precise sense, in a way that is compatible with the cartesian product.

We prove a congruence criterion for the algebraic theory of power operations in Morava E-theory, analogous to Wilkerson's congruence criterion for torsion free λ-rings. In addition, we provide a geometric description of this congruence criterion, in terms of sheaves on the moduli problem of deformations of formal groups and Frobenius isogenies.

In his 1965 paper C. T. C. Wall demonstrated that if a CW complex Y is finitely dominated, then the reduced projective class group of Y contains an obstruction which vanishes if and only if Y is homotopy equivalent to a finite CW complex. Wall also demonstrated that such an obstruction is invariant under homotopy equivalences. Subsequently Sum and Product Theorems for this obstruction were proved by L. C. Siebenmann.

In his second paper on the subject Wall gives an algebraic definition of the relative finiteness obstruction. If a CW complex Y is finitely dominated rel. a subcomplex X, then the reduced projective class group of Y contains an obstruction which vanishes if and only if Y is homotopy equivalent to a finite complex rel. X.

In this paper we will use a geometric construction to reduce the relative finiteness obstruction to the non-relative version. We will demonstrate that the relative finiteness obstruction is invariant under certain types of homotopy equivalences. We will also prove the relative versions of the Sum and the Product Theorems.