Mapping cone (topology) explained

In mathematics, especially homotopy theory, the mapping cone is a construction in topology analogous to a quotient space and denoted

C _{f} . Alternatively, it is also called the homotopy cofiber and also notated Cf . Its dual, a fibration, is called the mapping fiber. The mapping cone can be understood to be a mapping cylinder Mf with the initial end of the cylinder collapsed to a point. Mapping cones are frequently applied in the homotopy theory of pointed spaces.

Definition

f\colon X \to Y , the mapping cone C _{f} is defined to be the quotient space of the mapping cylinder (X x I) \sqcup _{f} Y with respect to the equivalence relation \forall x,x' \in X, (x, 0 ) \sim \left(x', 0 \right) , (x, 1 ) \sim f(x) . Here I denotes the unit interval [0, 1] with its standard topology. Note that some authors (like J. Peter May) use the opposite convention, switching 0 and 1.

Visually, one takes the cone on X (the cylinder

X x I with one end (the 0 end) collapsed to a point), and glues the other end onto Y via the map f (the 1 end).

Coarsely, one is taking the quotient space by the image of X, so

C _{f} = 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

f\colon (X, x _{0)} \to (Y, y _{0)} (so f\colon x _{0} \mapsto y _{0} ), one also identifies all of x _{0} x I . Formally, (x _{0,} t) \sim \left(x _{0,} t'\right) . Thus one end and the "seam" are all identified with y _{0.}

Example of circle

If

X is the circle S ^{1} , the mapping cone C _{f} can be considered as the quotient space of the disjoint union of Y with the disk D ^{2} formed by identifying each point x on the boundary of D ^{2} to the point f(x) in Y.

Consider, for example, the case where Y is the disk

D ^{2} , and f\colon S ^{1} \to Y = D ^{2} is the standard inclusion of the circle S ^{1} as the boundary of D ^{2} . Then the mapping cone C _{f} is homeomorphic to two disks joined on their boundary, which is topologically the sphere S ^{2} .

Double mapping cylinder

The mapping cone is a special case of the double mapping cylinder. This is basically a cylinder

X x I joined on one end to a space Y _{1} via a mapf _{1:} X \to Y _{1}

and joined on the other end to a space

Y _{2} via a mapf _{2:} X \to Y _{2}

The mapping cone is the degenerate case of the double mapping cylinder (also known as the homotopy pushout), in which one of

Y _{1,} Y _{2} is a single point.

Dual construction: the mapping fibre

F _{f} . Given the pointed map f\colon (X, x _{0)} \to (Y, y _{0),} one defines the mapping fiber as^[1] F _{f} = \left\{(x, \omega) \in X x Y ^{I} : \omega( 0 ) = y _{0} and \omega( 1 ) = f(x)\right\} .

Here, I is the unit interval and

\omega is a continuous path in the space (the exponential object) Y ^{I} . The mapping fiber is sometimes denoted as Mf ; however this conflicts with the same notation for the mapping cylinder.X x _{f} Y which is dual to the pushout X\sqcup _{f} Y used to construct the mapping cone.^[2] In this particular case, the duality is essentially that of currying, in that the mapping cone (X x I)\sqcup _{f} Y has the curried form X x _{f} (I\to Y) where I\to Y is simply an alternate notation for the space Y ^{I} of all continuous maps from the unit interval to Y . The two variants are related by an adjoint functor. Observe that the currying preserves the reduced nature of the maps: in the one case, to the tip of the cone, and in the other case, paths to the basepoint.

Applications

CW-complexes

Attaching a cell.

Effect on fundamental group

Given a space X and a loop

\alpha\colon S ^{1} \to X representing an element of the fundamental group of X, we can form the mapping cone C _{\alpha} . The effect of this is to make the loop \alpha contractible in C _{\alpha} , and therefore the equivalence class of \alpha in the fundamental group of C _{\alpha} will be simply the identity element.

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. Namely, if E is a homology theory, and

i\colon A \to X is a cofibration, thenE _{*(X,A)} = E _{*(X/A,*)} = \tilde E _{*(X/A)} ,

which follows by applying excision to the mapping cone.^[2]

Relation to homotopy (homology) equivalences

A map

f\colon X → Y between simply-connected CW complexes is a homotopy equivalence if and only 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.^[3]

Let

H _{*} be a fixed homology theory. The map f\colon X → Y induces isomorphisms on H _{*} , if and only if the map \{pt\}\hookrightarrow C _{f} induces an isomorphism on H _{*} , i.e., H _{*(C} _{f,pt)=0} .

Mapping cones are famously used to construct the long coexact Puppe sequences, from which long exact sequences of homotopy and relative homotopy groups can be obtained.^[1]

See also

• Cofibration
• Mapping cone (homological algebra)

References

Notes and References

1. Book: Rotman, Joseph J.. Joseph J. Rotman

   . Joseph J. Rotman. An Introduction to Algebraic Topology. 1988. Springer-Verlag . 0-387-96678-1. See Chapter 11 for proof.

2. Book: May, J. Peter. J. Peter May

   . J. Peter May. A Concise Course in Algebraic Topology. 1999. Chicago Lectures in Mathematics . 0-226-51183-9. See Chapter 6.

3. • Book: Hatcher, Allen. Allen Hatcher

   . Allen Hatcher. Algebraic topology. Cambridge University Press. Cambridge. 2002. 9780521795401.