Concept

Tour de Postnikov

In homotopy theory, a branch of algebraic topology, a Postnikov system (or Postnikov tower) is a way of decomposing a topological space's homotopy groups using an inverse system of topological spaces whose homotopy type at degree agrees with the truncated homotopy type of the original space . Postnikov systems were introduced by, and are named after, Mikhail Postnikov. A Postnikov system of a path-connected space is an inverse system of spaces with a sequence of maps compatible with the inverse system such that The map induces an isomorphism for every . for . Each map is a fibration, and so the fiber is an Eilenberg–MacLane space, . The first two conditions imply that is also a -space. More generally, if is -connected, then is a -space and all for are contractible. Note the third condition is only included optionally by some authors. Postnikov systems exist on connected CW complexes, and there is a weak homotopy-equivalence between and its inverse limit, so showing that is a CW approximation of its inverse limit. They can be constructed on a CW complex by iteratively killing off homotopy groups. If we have a map representing a homotopy class , we can take the along the boundary map , killing off the homotopy class. For this process can be repeated for all , giving a space which has vanishing homotopy groups . Using the fact that can be constructed from by killing off all homotopy maps , we obtain a map . One of the main properties of the Postnikov tower, which makes it so powerful to study while computing cohomology, is the fact the spaces are homotopic to a CW complex which differs from only by cells of dimension . The sequence of fibrations have homotopically defined invariants, meaning the homotopy classes of maps , give a well defined homotopy type . The homotopy class of comes from looking at the homotopy class of the classifying map for the fiber . The associated classifying map is hence the homotopy class is classified by a homotopy class called the n-th Postnikov invariant of , since the homotopy classes of maps to Eilenberg-Maclane spaces gives cohomology with coefficients in the associated abelian group.

À propos de ce résultat
Cette page est générée automatiquement et peut contenir des informations qui ne sont pas correctes, complètes, à jour ou pertinentes par rapport à votre recherche. Il en va de même pour toutes les autres pages de ce site. Veillez à vérifier les informations auprès des sources officielles de l'EPFL.

Graph Chatbot

Chattez avec Graph Search

Posez n’importe quelle question sur les cours, conférences, exercices, recherches, actualités, etc. de l’EPFL ou essayez les exemples de questions ci-dessous.

AVERTISSEMENT : Le chatbot Graph n'est pas programmé pour fournir des réponses explicites ou catégoriques à vos questions. Il transforme plutôt vos questions en demandes API qui sont distribuées aux différents services informatiques officiellement administrés par l'EPFL. Son but est uniquement de collecter et de recommander des références pertinentes à des contenus que vous pouvez explorer pour vous aider à répondre à vos questions.