We show that there is a homotopy cofiber sequence of spectra relating Carlsson’s deformation K-theory of a group G to its “deformation representation ring,” analogous to the Bott periodicity sequence relating connective K-theory to ordinary homology. We then apply this to study simultaneous similarity of unitary matrices. The algebraic K-theory of a category uses the machinery of infinite loop ...

In this paper we consider a construction in an arbitrary triangulated category T which resembles the notion of a Moore spectrum in algebraic topology. Namely, given a compact object C of T satisfying some finite tilting assumptions, we obtain a functor which “approximates” objects from the module category of the endomorphism algebra of C in T . This generalises and extends a construction of Jør...

In this paper we present several algorithms related with the computation of the homology of groups, from a geometric perspective (that is to say, carrying out the calculations by means of simplicial sets and using techniques of Algebraic Topology). More concretely, we have developed some algorithms which, making use of the effective homology method, construct the homology groups of Eilenberg-Ma...

Let $(K,v)$ be a valued field. We review some results of MacLane and Vaqui\'e on extensions $v$ to valuations the polynomial ring $K[x]$. introduce certain MacLane-Vaqui\'e chains residually transcendental valuations, we prove that every valuation $\mu$ $K[x]$ is limit finite or countably infinite chain. This chain underlying essentially unique contains arithmetic data yielding an explicit desc...

The planarity theorems of MacLane and Whitney are extended to compact graph–like spaces. This generalizes recent results of Bruhn and Stein (MacLane’s Theorem for the Freudenthal compactification of a locally finite graph) and of Bruhn and Diestel (Whitney’s Theorem for an identification space obtained from a graph in which no two vertices are joined by infinitely many edge-disjoint paths).

We prove that the Morava-K-theory-based Eilenberg-Moore spectral sequence has good convergence properties whenever the base space is a p-local finite Postnikov system with vanishing (n + 1)st homotopy group.

This paper arose from attempting to understand Bousfield localization functors in stable homotopy theory. All spectra will be p-local for a prime p throughout this paper. Recall that if E is a spectrum, a spectrum X is Eacyclic if E∧X is null. A spectrum is E-local if every map from an E-acyclic spectrum to it is null. A map X → Y is an E-equivalence if it induces an isomorphism on E∗, or equiv...

The problem of describing the homotopy groups of spheres has been fundamental to algebraic topology for around 80 years. There were periods when specific computations were important and periods when the emphasis favored theory. Many mathematical invariants have expressions in terms of homotopy groups, and at different times the subject has found itself located in geometric topology, algebra, al...

One can do the motivic homotopy theory in the context of different motivic homotopy categories. One can vary the topology on the category of schemes used to define the homotopy category or one can vary the category of schemes itself considering only schemes satisfying certain conditions. The category obtained by taking smooth schemes and the Nisnevich topology seems to play a distinguished role...

Introduction. Let £ = (£ , p, B, F) denote a two stage Postnikov system with stable fe-invariant. We announce results about H*(ÜE) as a Hopf algebra over the Steenrod algebra. Mod 2 cohomology is used exclusively. Unexplained notation is from [4] and [S]. 1 am grateful to D. Anderson, W. Massey, F . Peterson and H. Salomonsen for many useful remarks. We make the following assumptions on £, in a...