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[[File:Mapping cone.PNG|thumb|The mapping cone of <math>f: X \to Y</math> is obtained by gluing the cone over ''X'' to ''Y''.]] | |||
In [[mathematics]], especially [[homotopy theory]], the '''mapping cone''' is a construction <math>C_f</math> of [[topology]], analogous to a [[quotient space]]. It is also called the '''homotopy cofiber,''' and also notated <math>Cf</math>. | |||
==Definition== | |||
Given a [[continuous function|map]] <math>f\colon X \to Y</math>, the mapping cone <math>C_f</math> is defined to be the quotient topological space of <math>(X \times I) \sqcup Y</math> with respect to the equivalence relation <math>(x, 0) \sim (x',0)\,</math>, <math>(x,1) \sim f(x)\,</math> on ''X''. Here <math>I</math> 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 <math>X \times I</math> 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|homology of a pair]] and the notion of an [[n-connected|''n''-connected]] map. | |||
The above is the definition for a map of unpointed spaces; for a map of pointed spaces <math>f\colon (X,x_0) \to (Y,y_0),</math> (so <math>f\colon x_0 \mapsto y_0</math>), one also identifies all of <math>{x_0}\times I</math>; formally, <math>(x_0,t) \sim (x_0,t')\,.</math> Thus one end and the "seam" are all identified with <math>y_0.</math> | |||
=== Example of circle === | |||
If <math>X</math> is the [[circle]] ''S''<sup>1</sup>, ''C''<sub>''f''</sub> can be considered as the [[quotient space]] of the [[disjoint union]] of ''Y'' with the [[disk (mathematics)|disk]] ''D''<sup>2</sup> formed by identifying a point ''x'' on the [[boundary (topology)|boundary]] of ''D''<sup>2</sup> to the point ''f(x)'' in ''Y''. | |||
Consider, for example, the case where ''Y'' is the disc ''D''<sup>2</sup>, and | |||
:''f'': ''S''<sup>1</sup> → ''Y'' = ''D''<sup>2</sup> | |||
is the standard inclusion of the circle ''S''<sup>1</sup> as the boundary of ''D''<sup>2</sup>. Then the mapping cone ''C''<sub>''f''</sub> is [[homeomorphism|homeomorphic]] to two disks joined on their boundary, which is topologically the sphere ''S''<sup>2</sup>. | |||
=== Double mapping cylinder === | |||
The mapping cone is a special case of the double [[mapping cylinder]]. This is basically a cylinder joined on one end to a space ''X''<sub>1</sub> via the [[continuous function|map]] | |||
:''f''<sub>1</sub>: ''S''<sup>1</sup> → ''X''<sub>1</sub> | |||
and joined on the other end to a space ''X''<sub>2</sub> via the map | |||
:''f''<sub>2</sub>: S<sup>1</sup> → ''X''<sub>2</sub>. | |||
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. | |||
==Applications== | |||
===CW-complexes=== | |||
Attaching a cell | |||
===Effect on fundamental group=== | |||
Given a [[topological space|space]] ''X'' and a loop | |||
:<math>\alpha\colon S^1 \to X</math> | |||
representing an element of the [[fundamental group]] of ''X'', we can form the mapping cone ''C''<sub>''α''</sub>. The effect of this is to make the loop ''α'' [[contractible]] in ''C''<sub>''α''</sub>, and therefore the [[equivalence class]] of ''α'' in the fundamental group of ''C''<sub>''α''</sub> 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: | |||
If ''E'' is a [[homology theory]], and <math>i\colon A \to X</math> is a [[cofibration]], then <math>E_*(X,A) = E_*(X/A,*) = \tilde E_*(X/A)</math>, which follows by applying excision to the mapping cone.<ref> | |||
[http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf Peter May "A Concise Course in Algebraic Topology", section 14.2] | |||
</ref> | |||
==Relation to homotopy (homology) equivalences== | |||
A map <math> f\colon X\rightarrow Y</math> 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|''n''-connected]] (as a map) if its mapping cone is ''n''-connected (as a space), plus a little more. | |||
See A. Hatcher [http://www.math.cornell.edu/~hatcher/AT/ATpage.html Algebraic Topology].{{PN|date=February 2013}} | |||
Let <math> \mathbb{}H_*</math> be a fixed [[homology theory]]. The map <math> f:X\rightarrow Y</math> induces isomorphisms on <math>\mathbb{}H_*</math>, if and only if the map <math> \{\cdot\}\hookrightarrow C_f</math> induces an isomorphism on <math> \mathbb{}H_*</math>, i.e. <math> \mathbb{}H_*(C_f,pt)=0</math>. | |||
==See also== | |||
* [[Mapping cone (homological algebra)]] | |||
==References== | |||
<References /> | |||
[[Category:Algebraic topology]] |
Revision as of 07:14, 12 November 2013
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 .
Definition
Given a map , 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
Example of circle
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 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
- f: S1 → Y = 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 cylinder
The mapping cone is a special case of the double mapping cylinder. This is basically a cylinder joined on one end to a space X1 via the map
- 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.
Applications
CW-complexes
Attaching a cell
Effect on fundamental group
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 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, and is a cofibration, then , which follows by applying excision to the mapping cone.[1]
Relation to homotopy (homology) equivalences
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. See A. Hatcher Algebraic Topology.Template:PN
Let be a fixed homology theory. The map induces isomorphisms on , if and only if the map induces an isomorphism on , i.e. .