Êtes-vous un étudiant de l'EPFL à la recherche d'un projet de semestre?
Travaillez avec nous sur des projets en science des données et en visualisation, et déployez votre projet sous forme d'application sur Graph Search.
Concept
Suite de Puppe
Résumé
In mathematics, the Puppe sequence is a construction of homotopy theory, so named after Dieter Puppe. It comes in two forms: a long exact sequence, built from the mapping fibre (a fibration), and a long coexact sequence, built from the mapping cone (which is a cofibration). Intuitively, the Puppe sequence allows us to think of homology theory as a functor that takes spaces to long-exact sequences of groups. It is also useful as a tool to build long exact sequences of relative homotopy groups.
Let be a continuous map between pointed spaces and let denote the mapping fibre (the fibration dual to the mapping cone). One then obtains an exact sequence:
where the mapping fibre is defined as:
Observe that the loop space injects into the mapping fibre: , as it consists of those maps that both start and end at the basepoint . One may then show that the above sequence extends to the longer sequence
The construction can then be iterated to obtain the exact Puppe sequence
The exact sequence is often more convenient than the coexact sequence in practical applications, as Joseph J. Rotman explains:
(the) various constructions (of the coexact sequence) involve quotient spaces instead of subspaces, and so all maps and homotopies require more scrutiny to ensure that they are well-defined and continuous.
As a special case, one may take X to be a subspace A of Y that contains the basepoint y0, and f to be the inclusion of A into Y. One then obtains an exact sequence in the :
where the are the homotopy groups, is the zero-sphere (i.e. two points) and denotes the homotopy equivalence of maps from U to W. Note that . One may then show that
is in bijection to the relative homotopy group , thus giving rise to the relative homotopy sequence of pairs
The object is a group for and is abelian for .
As a special case, one may take f to be a fibration . Then the mapping fiber Mp has the homotopy lifting property and it follows that Mp and the fiber have the same homotopy type.
Source officielle
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.
Proximité ontologique