3 Motivic cohomology and algebraic cobordisms. 21 3.1 A topological lemma. . . . . . . . . . . . . . . . . . . . . . . . 21 3.2 Homotopy categories of algebraic varieties . . . . . . . . . . . 25 3.3 Eilenberg-MacLane spectra and motivic cohomology. . . . . . . 30 3.4 Topological realization functor. . . . . . . . . . . . . . . . . . . 33 3.5 Algebraic cobordisms. . . . . . . . . . . . . . . . ...

k — a regular commutative ring. (Main case of interest, as we shall see, is k = Z.) kG — the group ring over k of a group G. BG — the classifying space of G, a space with contractible universal cover and fundamental group G. Unique up to homotopy equivalence. Has the property that H∗(BG) is the Eilenberg-MacLane homology of G. K(R) — the (non-connective) K-theory spectrum of a ring R. If one ig...

We introduce a notion of ‘cover of level n’ for a topological space, or more generally any Grothendieck site, with the key property that simplicial homotopy classes computed along the filtered diagram of n-covers biject with global homotopy classes when the target is an n-type. When the target is an Eilenberg–MacLane sheaf, this specializes to computing derived functor cohomology, up to degree ...

Let Tn be the kernel of the natural map Out(Fn) → GLn(Z). We use combinatorial Morse theory to prove that Tn has an Eilenberg–MacLane space which is (2n − 4)-dimensional and that H2n−4(Tn, Z) is not finitely generated (n ≥ 3). In particular, this recovers the result of Krstić–McCool that T3 is not finitely presented. We also give a new proof of the fact, due to Magnus, that Tn is finitely gener...

We discuss the Bousfield localization LEX for any spectrum E and any HR-module X, where R is a ring with unit. Due to the splitting property of HR-modules, it is enough to study the localization of Eilenberg– MacLane spectra. Using general results about stable f -localizations, we give a method to compute the localization of an Eilenberg–MacLane spectrum LEHG for any spectrum E and any abelian ...

Given any group Γ, there is an aspherical CW -complex BΓ (also denoted by K(Γ, 1)) with fundamental group Γ; moreover, BΓ is unique up to homotopy equivalence (cf. [Hu]). BΓ is called the classifying space of Γ. (BΓ is also called an Eilenberg-MacLane space for Γ.) So, the theory of aspherical CW -complexes, up to homotopy, is identical with the theory of groups. This point of view led to the n...

In this paper, using a relation between Schur multipliers of pairs and triples of groups, the fundamental group and homology groups of a homotopy pushout of Eilenberg-MacLane spaces, we present among other things some behaviors of Schur multipliers of pairs and triples with respect to free, amalgamated free, and direct products and also direct limits of groups with topological approach.