In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Puppe sequence is a construction of
homotopy theory
In 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 topolog ...
, so named after
. It comes in two forms: a
long exact sequence
An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next.
Definition
In the context ...
, built from the
mapping fibre (a
fibration
The 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 map ...
), and a long coexact sequence, built from the
mapping cone Mapping cone may refer to one of the following two different but related concepts in mathematics:
* Mapping cone (topology)
* Mapping cone (homological algebra)
In homological algebra, the mapping cone is a construction on a map of chain complexe ...
(which is a
cofibration In mathematics, in particular homotopy theory, a continuous mapping
:i: A \to X,
where A and X are topological spaces, is a cofibration if it lets homotopy classes of maps ,S/math> be extended to homotopy classes of maps ,S/math> whenever a map ...
).
[ Joseph J. Rotman, ''An Introduction to Algebraic Topology'' (1988) Springer-Verlag ''(See Chapter 11 for construction.)''] Intuitively, the Puppe sequence allows us to think of
homology theory
In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topolog ...
as a
functor
In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...
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.
Exact Puppe sequence
Let
be a
continuous map
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value ...
between
pointed space
In mathematics, a pointed space or based space is a topological space with a distinguished point, the basepoint. The distinguished point is just simply one particular point, picked out from the space, and given a name, such as x_0, that remains u ...
s and let
denote the
mapping fibre (the
fibration
The 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 map ...
dual to the
mapping cone Mapping cone may refer to one of the following two different but related concepts in mathematics:
* Mapping cone (topology)
* Mapping cone (homological algebra)
In homological algebra, the mapping cone is a construction on a map of chain complexe ...
). One then obtains an exact sequence:
:
where the mapping fibre is defined as:
[
:
Observe that the ]loop space
In topology, a branch of mathematics, the loop space Ω''X'' of a pointed topological space ''X'' is the space of (based) loops in ''X'', i.e. continuous pointed maps from the pointed circle ''S''1 to ''X'', equipped with the compact-open topology ...
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.''
]
Examples
Example: Relative homotopy
As a special case,[ one may take ''X'' to be a subspace ''A'' of ''Y'' that contains the basepoint ''y''0, and ''f'' to be the inclusion of ''A'' into ''Y''. One then obtains an exact sequence in the ]category of pointed spaces
In mathematics, a pointed space or based space is a topological space with a distinguished point, the basepoint. The distinguished point is just simply one particular point, picked out from the space, and given a name, such as x_0, that remains u ...
:
:
where the are the homotopy group
In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted \pi_1(X), which records information about loops in a space. Intuitively, homotop ...
s, is the zero-sphere (i.e. two points) and