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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 ...
, specifically
algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...
, the mapping cylinder of a
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ...
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
f between
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
s X and Y is the
quotient In arithmetic, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a ...
:M_f = (( ,1times X) \amalg Y)\,/\,\sim where the \amalg denotes the
disjoint union In mathematics, a disjoint union (or discriminated union) of a family of sets (A_i : i\in I) is a set A, often denoted by \bigsqcup_ A_i, with an injection of each A_i into A, such that the images of these injections form a partition of A (th ...
, and ∼ is the
equivalence relation In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relation ...
generated by :(0,x)\sim f(x)\quad\textx\in X. That is, the mapping cylinder M_f is obtained by gluing one end of X\times ,1/math> to Y via the map f. Notice that the "top" of the cylinder \\times X is
homeomorphic In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphi ...
to X, while the "bottom" is the space f(X)\subset Y. It is common to write Mf for M_f, and to use the notation \sqcup_f or \cup_f for the mapping cylinder construction. That is, one writes :Mf = ( ,1times X) \cup_f Y with the subscripted cup symbol denoting the equivalence. The mapping cylinder is commonly used to construct 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 ...
Cf, obtained by collapsing one end of the cylinder to a point. Mapping cylinders are central to the definition of
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 ...
s.


Basic properties

The bottom ''Y'' is a
deformation retract In topology, a branch of mathematics, a retraction is a continuous mapping from a topological space into a subspace that preserves the position of all points in that subspace. The subspace is then called a retract of the original space. A deforma ...
of M_f. The projection M_f \to Y splits (via Y \ni y \mapsto y \in Y \subset M_f), and the deformation retraction R is given by: : R: M_f \times I \rightarrow M_f :( ,xs) \mapsto \cdot t,x (y, s) \mapsto y (where points in Y stay fixed because ,x \cdot 0,x/math> for all s). The map f:X \to Y is a
homotopy equivalence In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deforma ...
if and only if the "top" \\times X is a strong deformation retract of M_f. An explicit formula for the strong deformation retraction can be worked out.


Examples


Mapping cylinder of a fiber bundle

For a
fiber bundle In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a p ...
\pi:P \to X with fiber F, the mapping cylinder :M_\pi = (( ,1times P) \coprod X)/\sim has the equivalence relation : (0,p_) \sim (0, q_) for p_,q_ \in F_x. Then, there is a canonical map sending a point , p_, x\in M_\pi to the point x \in X, giving a fiber bundle :p:M_\pi \to X whose fiber is the cone CF. To see this, notice the fiber over a point x\in X is the quotient space : ,1times P \coprod \/\sim where every point in \\times P is equivalent.


Interpretation

The mapping cylinder may be viewed as a way to replace an arbitrary map by an equivalent
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 ...
, in the following sense: Given a map f\colon X \to Y, the mapping cylinder is a space M_f, ''together with'' a cofibration \tilde f\colon X \to M_f and a
surjective In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of i ...
homotopy equivalence In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deforma ...
M_f \to Y (indeed, ''Y'' is a
deformation retract In topology, a branch of mathematics, a retraction is a continuous mapping from a topological space into a subspace that preserves the position of all points in that subspace. The subspace is then called a retract of the original space. A deforma ...
of M_f), such that the
composition Composition or Compositions may refer to: Arts and literature *Composition (dance), practice and teaching of choreography *Composition (language), in literature and rhetoric, producing a work in spoken tradition and written discourse, to include v ...
X \to M_f \to Y equals ''f''. Thus the space ''Y'' gets replaced with a homotopy equivalent space M_f, and the map ''f'' with a lifted map \tilde f. Equivalently, the diagram :f\colon X \to Y gets replaced with a diagram :\tilde f\colon X \to M_f together with a homotopy equivalence between them. The construction serves to replace any map of topological spaces by a homotopy equivalent cofibration. Note that pointwise, 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 ...
is a closed
inclusion Inclusion or Include may refer to: Sociology * Social inclusion, aims to create an environment that supports equal opportunity for individuals and groups that form a society. ** Inclusion (disability rights), promotion of people with disabiliti ...
.


Applications

Mapping cylinders are quite common homotopical tools. One use of mapping cylinders is to apply theorems concerning inclusions of spaces to general maps, which might not be
injective In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositiv ...
. Consequently, theorems or techniques (such as
homology Homology may refer to: Sciences Biology *Homology (biology), any characteristic of biological organisms that is derived from a common ancestor * Sequence homology, biological homology between DNA, RNA, or protein sequences *Homologous chrom ...
,
cohomology In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...
or
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 ...
) which are only dependent on the homotopy class of spaces and maps involved may be applied to f\colon X\rightarrow Y with the assumption that X \subset Y and that f is actually the inclusion of a subspace. Another, more intuitive appeal of the construction is that it accords with the usual mental image of a function as "sending" points of X to points of Y, and hence of embedding X within Y, despite the fact that the function need not be one-to-one.


Categorical application and interpretation

One can use the mapping cylinder to construct
homotopy colimit In mathematics, especially in algebraic topology, the homotopy limit and colimitpg 52 are variants of the notions of limit and colimit extended to the homotopy category \text(\textbf). The main idea is this: if we have a diagramF: I \to \textbfcon ...
s: this follows from the general statement that any
category Category, plural categories, may refer to: Philosophy and general uses * Categorization, categories in cognitive science, information science and generally *Category of being * ''Categories'' (Aristotle) *Category (Kant) *Categories (Peirce) * ...
with all
pushout A ''pushout'' is a student who leaves their school before graduation, through the encouragement of the school. A student who leaves of their own accord (e.g., to work or care for a child), rather than through the action of the school, is consider ...
s and
coequalizer In category theory, a coequalizer (or coequaliser) is a generalization of a quotient by an equivalence relation to objects in an arbitrary category. It is the categorical construction dual to the equalizer. Definition A coequalizer is a co ...
s has all
colimit In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products, pullbacks and inverse limits. The dual notion of a colimit generalizes constructions such ...
s. That is, given a diagram, replace the maps by cofibrations (using the mapping cylinder) and then take the ordinary pointwise limit (one must take a bit more care, but mapping cylinders are a component). Conversely, the mapping cylinder is the
homotopy pushout In mathematics, especially in algebraic topology, the homotopy limit and colimitpg 52 are variants of the notions of limit and colimit extended to the homotopy category \text(\textbf). The main idea is this: if we have a diagramF: I \to \textbfc ...
of the diagram where f\colon X \to Y and \text_X\colon X \to X.


Mapping telescope

Given a
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
of maps :X_1 \xrightarrow X_2 \xrightarrow X_3 \to\cdots the mapping telescope is the homotopical
direct limit In mathematics, a direct limit is a way to construct a (typically large) object from many (typically smaller) objects that are put together in a specific way. These objects may be groups, rings, vector spaces or in general objects from any categor ...
. If the maps are all already cofibrations (such as for the
orthogonal group In mathematics, the orthogonal group in dimension , denoted , is the Group (mathematics), group of isometry, distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by ...
s O(n) \subset O(n+1)), then the direct limit is the union, but in general one must use the mapping telescope. The mapping telescope is a sequence of mapping cylinders, joined end-to-end. The picture of the construction looks like a stack of increasingly large cylinders, like a telescope. Formally, one defines it as :\Bigl(\coprod_i ,1\times X_i\Bigr) / ((0,x_i) \sim (1,f_i(x_i))).


See also

*
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 ...
* Mapping cylinder (homological algebra) *
Homotopy colimit In mathematics, especially in algebraic topology, the homotopy limit and colimitpg 52 are variants of the notions of limit and colimit extended to the homotopy category \text(\textbf). The main idea is this: if we have a diagramF: I \to \textbfcon ...
* Mapping path space, which can be viewed as the mapping cocylinder


References

* {{DEFAULTSORT:Mapping Cylinder Algebraic topology