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Mapping cone (topology)

An illustration of a mapping cone; that is, a cone is glued to a space along some function .

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.

DefinitionEdit

Given a map  , the mapping cone   is defined to be the quotient topological space of the mapping cylinder   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  

Example of circleEdit

If   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 each 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

f: S1Y = D2

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.

Double mapping cylinderEdit

The mapping cone is a special case of the double mapping cylinder. This is basically a cylinder   joined on one end to a space   via a map

 

and joined on the other end to a space   via a map

 

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

Dual construction: the mapping fibreEdit

The dual to the mapping cone is the mapping fibre  . Given the pointed map  , one defines the mapping fiber as[1]

 

Here, I is the unit interval and   is a continuous path in the space (the exponential object)  . The mapping fiber is sometimes denoted as  ; however this conflicts with the same notation for the mapping cylinder.

It is dual to the mapping cone in the sense that the product above is essentially the fibered product or pullback   which is dual to the pushout   used to construct the mapping cone.[2] In this particular case, the duality is essentially that of currying, in that the mapping cone   has the curried form   where   is simply an alternate notation for the space   of all continuous maps from the unit interval to  . 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.

ApplicationsEdit

CW-complexesEdit

Attaching a cell

Effect on fundamental groupEdit

Given a space 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.

Homology of a pairEdit

The mapping cone lets one interpret the homology of a pair as the reduced homology of the quotient:

If E is a homology theory, and   is a cofibration, then  , which follows by applying excision to the mapping cone.[2]

Relation to homotopy (homology) equivalencesEdit

A map   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][page needed]

Let   be a fixed homology theory. The map   induces isomorphisms on  , if and only if the map   induces an isomorphism on  , i.e.  .

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 alsoEdit

ReferencesEdit

  1. ^ a b Joseph J. Rotman, An Introduction to Algebraic Topology (1988) Springer-Verlag ISBN 0-387-96678-1 (See Chapter 11 for proof.)
  2. ^ a b J.P. May, A Concise Course in Algebraic Topology, (1999) Chicago Lectures in Mathematics ISBN 0-226-51183-9 (See Chapter 6.)
  3. ^ A. Hatcher, Algebraic Topology.