Let be a field. We compute the set ofnaivehomotopy classes of pointed -scheme endomorphisms of the projective line . Our result compares well with Morel’s computation in [11] of thegroup of -homotopy classes of pointed endomorphisms of : the set admits an a priori monoid structure such that the canonical map is a group completion.
For a prime , we compute the algebraic -theory modulo and of the mod Adams summand, using topological cyclic homology. On the way, we evaluate its modulo and topological Hochschild homology. Using a localization sequence, we also compute the -theory modulo and of the first Morava -theory.
Because of its strong interaction with almost every part of pure mathematics, algebraic K-theory has had a spectacular development since its origin in the late fifties. The objective of this paper is to provide the basic definitions of the algebraic K-theory of rings and an overview of the main classical theorems. Since the algebraic K-groups of a ring R are the homotopy groups of a topological space associated with the general linear group over R, it is obvious that many general results follow...
In questo articolo si riassumono le definizioni e le principali proprietà dei gruppi di ostruzione con decorazione di tipo LS e LP. Si stabiliscono nuove relazioni fra questi gruppi e si descrivono le proprietà delle mappe naturali fra differenti gruppi con decorazione. Si costruiscono varie successioni spettrali, contenenti questi gruppi con decorazione, e si studiano la loro connessione con le successioni spettrali in -teoria per certe estensioni quadratiche di antistrutture. Infine, si introduce...
A countable CW complex K is quasi-finite (as defined by A. Karasev) if for every finite subcomplex M of K there is a finite subcomplex e(M) such that any map f: A → M, where A is closed in a separable metric space X satisfying XτK, has an extension g: X → e(M). Levin's results imply that none of the Eilenberg-MacLane spaces K(G,2) is quasi-finite if G ≠ 0. In this paper we discuss quasi-finiteness of all Eilenberg-MacLane spaces. More generally, we deal with CW complexes with finitely many...
The following two homotopic notions are important in many domains of differential geometry: - homotopic homomorphisms between principal bundles (and between other objects), - homotopic subbundles. They play a role, for example, in many fundamental problems of characteristic classes. It turns out that both these notions can be - in a natural way - expressed in the language of Lie algebroids. Moreover, the characteristic homomorphisms of principal bundles (the Chern-Weil homomorphism [K4], or the...
In this paper and its sequel we present a method that, under loose restrictions, is algorithmic for calculating the Nielsen type numbers NΦₙ(f) and NPₙ(f) of self maps f of hyperbolic surfaces with boundary and also of bouquets of circles. Because self maps of these surfaces have the same homotopy type as maps on wedges of circles, and the Nielsen periodic numbers are homotopy type invariant, we need concentrate only on the latter spaces. Of course the results will then automatically apply...