Mapping cone
Encyclopedia
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
.
, the mapping cone 
is defined to be the quotient topological space of 
with respect to the equivalence relation 
, 
on X. Here 
denotes the unit interval [0,1] with its standard topology. Note that some (like May) use the opposite convention, switching 0 and 1.
Visually, one takes the cone on X (the cylinder
with one end (the 0 end) identified to a point), and glues the other end onto Y via the map f (the identification of the 1 end).
Coarsely, one is taking the quotient space
by the image of X, so Cf "=" Y/f(X); this is not precisely correct because of point-set issues, but is the philosophy, and is made precise by such results as the homology of a pair and the notion of an n-connected
map.
The above is the definition for a map of unpointed spaces; for a map of pointed spaces
(so 
), one also identifies all of 
; formally, 
Thus one end and the "seam" are all identified with 
is the circle
S1, Cf can be considered as the quotient space
of the disjoint union
of Y with the disk
D2 formed by identifying a point x on the boundary
of D2 to the point f(x) in Y.
Consider, for example, the case where Y is the disc D2, and
is the standard inclusion of the circle S1 as the boundary of D2. Then the mapping cone Cf is homeomorphic
to two disks joined on their boundary, which is topologically the sphere S2.
. This is basically a cylinder joined on one end to a space X1 via the map
and joined on the other end to a space X2 via the map
The mapping cone is the degenerate case of the double mapping cylinder (also known as the homotopy pushout), in which one space is a single point.
X and a loop
representing an element of the fundamental group
of X, we can form the mapping cone Cα. The effect of this is to make the loop α contractible in Cα, and therefore the equivalence class of α in the fundamental group of Cα will be simply the identity element
.
Given a group presentation by generators and relations, one gets a 2-complex with that fundamental group.
If E is a homology theory
, and
is a cofibration
, then
, which follows by applying excision to the mapping cone.
between simply-connected CW complexes is a homotopy equivalence if its mapping cone is contractible.
More generally, a map is called n-connected
(as a map) if its mapping cone is n-connected (as a space), plus a little more.
See A. Hatcher Algebraic Topology.
Let
be a fixed homology theory
. The map
induces isomorphisms on 
, if and only if the map 
induces an isomorphism on 
, i.e. 
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, especially homotopy theory, the mapping cone is a construction
of topologyTopology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
, analogous to a quotient space
Quotient space
In topology and related areas of mathematics, a quotient space is, intuitively speaking, the result of identifying or "gluing together" certain points of a given space. The points to be identified are specified by an equivalence relation...
. It is also called the homotopy cofiber, and also notated
.Definition
Given a mapContinuous function
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
, the mapping cone is defined to be the quotient topological space of with respect to the equivalence relation , on X. Here denotes the unit interval [0,1] with its standard topology. Note that some (like May) use the opposite convention, switching 0 and 1.Visually, one takes the cone on X (the cylinder
with one end (the 0 end) identified to a point), and glues the other end onto Y via the map f (the identification of the 1 end).Coarsely, one is taking the quotient space
Quotient space
In topology and related areas of mathematics, a quotient space is, intuitively speaking, the result of identifying or "gluing together" certain points of a given space. The points to be identified are specified by an equivalence relation...
by the image of X, so Cf "=" Y/f(X); this is not precisely correct because of point-set issues, but is the philosophy, and is made precise by such results as the homology of a pair and the notion of an n-connected
N-connected
In the mathematical branch of algebraic topology, specifically homotopy theory, n-connectedness is a way to say that a space vanishes or that a map is an isomorphism "up to dimension n, in homotopy".-n-connected space:...
map.
The above is the definition for a map of unpointed spaces; for a map of pointed spaces
(so ), one also identifies all of ; formally, Thus one end and the "seam" are all identified with Example of circle
If is the circleCircle
A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....
S1, Cf can be considered as the quotient space
Quotient space
In topology and related areas of mathematics, a quotient space is, intuitively speaking, the result of identifying or "gluing together" certain points of a given space. The points to be identified are specified by an equivalence relation...
of the disjoint union
Disjoint union
In mathematics, the term disjoint union may refer to one of two different concepts:* In set theory, a disjoint union is a modified union operation that indexes the elements according to which set they originated in; disjoint sets have no element in common.* In probability theory , a disjoint union...
of Y with the disk
Disk (mathematics)
In geometry, a disk is the region in a plane bounded by a circle.A disk is said to be closed or open according to whether or not it contains the circle that constitutes its boundary...
D2 formed by identifying a point x on the boundary
Boundary (topology)
In topology and mathematics in general, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More precisely, it is the set of points in the closure of S, not belonging to the interior of S. An element of the boundary...
of D2 to the point f(x) in Y.
Consider, for example, the case where Y is the disc D2, and
- f: S1 → Y = D2
is the standard inclusion of the circle S1 as the boundary of D2. Then the mapping cone Cf is homeomorphic
Homeomorphism
In the mathematical field of topology, a homeomorphism or topological isomorphism or bicontinuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are...
to two disks joined on their boundary, which is topologically the sphere S2.
Double mapping cylinder
The mapping cone is a special case of the double mapping cylinderMapping cylinder
In mathematics, specifically algebraic topology, the mapping cylinder of a function f between topological spaces X and Y is the quotientM_f = \,/\,\sim...
. This is basically a cylinder joined on one end to a space X1 via the map
Continuous function
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
- f1: S1 → X1
and joined on the other end to a space X2 via the map
- f2: S1 → X2.
The mapping cone is the degenerate case of the double mapping cylinder (also known as the homotopy pushout), in which one space is a single point.
Effect on fundamental group
Given a spaceTopological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
X and a loop
representing an element of the fundamental group
Fundamental group
In mathematics, more specifically algebraic topology, the fundamental group is a group associated to any given pointed topological space that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other...
of X, we can form the mapping cone Cα. The effect of this is to make the loop α contractible in Cα, and therefore the equivalence class of α in the fundamental group of Cα will be simply the identity element
Identity element
In mathematics, an identity element is a special type of element of a set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them...
.
Given a group presentation by generators and relations, one gets a 2-complex with that fundamental group.
Homology of a pair
The mapping cone lets one interpret the homology of a pair as the reduced homology of the quotient:If E is a homology theory
Homology theory
In mathematics, homology theory is the axiomatic study of the intuitive geometric idea of homology of cycles on topological spaces. It can be broadly defined as the study of homology theories on topological spaces.-The general idea:...
, and
is a cofibrationCofibration
In mathematics, in particular homotopy theory, a continuous mappingi\colon A \to X,where A and X are topological spaces, is a cofibration if it satisfies the homotopy extension property with respect to all spaces Y. The name is because the dual condition, the homotopy lifting property, defines...
, then
, which follows by applying excision to the mapping cone.Relation to homotopy (homology) equivalences
A map between simply-connected CW complexes is a homotopy equivalence if its mapping cone is contractible.More generally, a map is called n-connected
N-connected
In the mathematical branch of algebraic topology, specifically homotopy theory, n-connectedness is a way to say that a space vanishes or that a map is an isomorphism "up to dimension n, in homotopy".-n-connected space:...
(as a map) if its mapping cone is n-connected (as a space), plus a little more.
See A. Hatcher Algebraic Topology.
Let
be a fixed homology theoryHomology theory
In mathematics, homology theory is the axiomatic study of the intuitive geometric idea of homology of cycles on topological spaces. It can be broadly defined as the study of homology theories on topological spaces.-The general idea:...
. The map
induces isomorphisms on , if and only if the map induces an isomorphism on , i.e. .