Postnikov systemIn 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 .
Mapping cylinderIn mathematics, specifically algebraic topology, the mapping cylinder of a continuous function between topological spaces and is the quotient where the denotes the disjoint union, and ∼ is the equivalence relation generated by That is, the mapping cylinder is obtained by gluing one end of to via the map . Notice that the "top" of the cylinder is homeomorphic to , while the "bottom" is the space . It is common to write for , and to use the notation or for the mapping cylinder construction.
Mapping cone (topology)In mathematics, especially homotopy theory, the mapping cone is a construction of topology, analogous to a quotient space. It is also called the homotopy cofiber, and also notated . Its dual, a fibration, is called the mapping fibre. The mapping cone can be understood to be a mapping cylinder , with one end of the cylinder collapsed to a point. Thus, mapping cones are frequently applied in the homotopy theory of pointed spaces. Given a map , the mapping cone is defined to be the quotient space of the mapping cylinder with respect to the equivalence relation , .
Puppe sequenceIn 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.
Path space fibrationIn algebraic topology, the path space fibration over a based space is a fibration of the form where is the path space of X; i.e., equipped with the compact-open topology. is the fiber of over the base point of X; thus it is the loop space of X. The space consists of all maps from I to X that may not preserve the base points; it is called the free path space of X and the fibration given by, say, , is called the free path space fibration. The path space fibration can be understood to be dual to the mapping cone.
FibrationThe notion of a fibration generalizes the notion of a fiber bundle and plays an important role in algebraic topology, a branch of mathematics. Fibrations are used, for example, in Postnikov systems or obstruction theory. In this article, all mappings are continuous mappings between topological spaces. A mapping satisfies the homotopy lifting property for a space if: for every homotopy and for every mapping (also called lift) lifting (i.e. ) there exists a (not necessarily unique) homotopy lifting (i.e.
Weak equivalence (homotopy theory)In mathematics, a weak equivalence is a notion from homotopy theory that in some sense identifies objects that have the same "shape". This notion is formalized in the axiomatic definition of a . A model category is a with classes of morphisms called weak equivalences, fibrations, and cofibrations, satisfying several axioms. The associated of a model category has the same objects, but the morphisms are changed in order to make the weak equivalences into isomorphisms.
CofibrationIn mathematics, in particular homotopy theory, a continuous mapping between topological spaces is a cofibration if it has the homotopy extension property with respect to all topological spaces . That is, is a cofibration if for each topological space , and for any continuous maps and with , for any homotopy from to , there is a continuous map and a homotopy from to such that for all and . (Here, denotes the unit interval .
Homotopy theoryIn mathematics, homotopy theory is a systematic study of situations in which maps can come with homotopies between them. It originated as a topic in algebraic topology but nowadays is studied as an independent discipline. Besides algebraic topology, the theory has also been used in other areas of mathematics such as algebraic geometry (e.g., A1 homotopy theory) and (specifically the study of ). In homotopy theory and algebraic topology, the word "space" denotes a topological space.
Homotopy groupIn mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted which records information about loops in a space. Intuitively, homotopy groups record information about the basic shape, or holes, of a topological space. To define the n-th homotopy group, the base-point-preserving maps from an n-dimensional sphere (with base point) into a given space (with base point) are collected into equivalence classes, called homotopy classes.
