Double mapping cylinder: Difference between revisions

Latest revision as of 22:44, 11 February 2014

This article describes a construct that involves some variant of taking a product of a topological space with the unit interval and then making some identifications, typically at the endpoints, based on some specific maps.
View more such constructs

Definition

Suppose $X,Y,Z$ are topological spaces and $f:X\to Z$ and $g:X\to Z$ are continuous maps. The double mapping cylinder of $f$ and $g$ is defined as the quotient of $X\times [0,1]\sqcup Y\sqcup Z$ via the relations $(x,0)\simeq f(x)$ and $(x,1)\simeq g(x)$ .

More specific constructions

Specific cases of the above arise either by setting $X=Y$ and $f$ the identity map (or correspondingly for $Z$ and $g$ ) or setting $Y$ or $Z$ to be a one-point space. If we impose only one constraint, the resultant construction is the construction corresponding to the other unspecified map. If we impose two constraints, then the resulting construction depends only on the input space $X$ .

Note that the roles of $Y$ and $Z$ can be interchanged here.

Name of construction	One-point spaces?	Identity maps?	Remaining input?	Conclusion
mapping cylinder	--	$f$ , so $X=Y$	$g:X\to Z$	With this stipulation, the double mapping cylinder is the same as the mapping cylinder for $g:X\to Z$ .
mapping cone	$Z$ , so $g$ sends all of $X$ to one point	--	$f:X\to Y$	With this stipulation, the double mapping cylinder is the same as the mapping cone for $f:X\to Y$ .
cone space	$Z$ , so $g$ sends all of $X$ to one point	$f$ , so $X=Y$	Only the space $X$	With this stipulation, we get the cone space for $X$
suspension	both $Y$ and $Z$	--	Only the space $X$	With this stipulation, we get the suspension of $X$
cylinder	--	both $f$ and $g$	only the space $X$	With this stipulation, we get the cylinder on $X$ , i.e., the product of $X$ and the unit interval

The join

The join of two spaces $A$ and $B$ can be constructed as a double mapping cylinder as follow: Set $X=A\times B$ , $Y=A$ and $Z=B$ , and let $f,g$ be the coordinate projection maps.

Generalizations

Mapping torus

Related notions

Mapping telescope

Facts

There is a relation between the homology of the double mapping cylinder of $f$ and $g$ , and the homologies of the spaces $X$ , $Y$ and $Z$ . The relation is described by the exact sequence for double mapping cylinder.

@@ Line 1: / Line 1: @@
+{{interval-cum-mapping construct}}
 ==Definition==
-Suppose <math>X,Y,Z</math> are [[topological space]]s and <math>f:X \to Z</math> and <math>g: Y \to Z</math> are [[continuous map]]s. The '''double mapping cylinder''' of <math>f</math> and <math>g</math> is defined as the quotient of <math>X \times [0,1] \sqcup Y \sqcup Z</math> via the relations <math>(x,0) \simeq f(x)</math> and <math>(x,1) \simeq g(x)</math>.
+Suppose <math>X,Y,Z</math> are [[topological space]]s and <math>f:X \to Z</math> and <math>g: X \to Z</math> are [[continuous map]]s. The '''double mapping cylinder''' of <math>f</math> and <math>g</math> is defined as the quotient of <math>X \times [0,1] \sqcup Y \sqcup Z</math> via the relations <math>(x,0) \simeq f(x)</math> and <math>(x,1) \simeq g(x)</math>.
+==More specific constructions==
+Specific cases of the above arise either by setting <math>X = Y</math> and <math>f</math> the identity map (or correspondingly for <math>Z</math> and <math>g</math>) or setting <math>Y</matH> or <math>Z</math> to be a [[one-point space]]. If we impose only one constraint, the resultant construction is the construction corresponding to the ''other'' unspecified map. If we impose two constraints, then the resulting construction depends only on the input space <math>X</math>.
+Note that the roles of <math>Y</math> and <math>Z</math> can be interchanged here.
+{| class="sortable" border="1"
+! Name of construction !! One-point spaces? !! Identity maps? !! Remaining input? || Conclusion
+|-
+| [[generalization of::mapping cylinder]] || -- || <math>f</math>, so <math>X = Y</math> || <math>g:X \to Z</math> || With this stipulation, the double mapping cylinder is the same as the mapping cylinder for <math>g:X \to Z</math>.
+|-
+| [[generalization of::mapping cone]] || <math>Z</math>, so <math>g</math> sends all of <math>X</math> to one point || -- || <math>f:X \to Y</math> ||With this stipulation, the double mapping cylinder is the same as the mapping cone for <math>f:X \to Y</math>.
+|-
+| [[generalization of::cone space]] || <math>Z</math>, so <math>g</math> sends all of <math>X</math> to one point || <math>f</math>, so <math>X = Y</math> || Only the space <math>X</math> || With this stipulation, we get the cone space for <math>X</math>
+|-
+| [[generalization of::suspension]] || both <math>Y</math> and <math>Z</math> || -- || Only the space <math>X</math> || With this stipulation, we get the suspension of <math>X</math>
+|-
+| [[generalization of::cylinder]] || -- || both <math>f</math> and <math>g</math> || only the space <math>X</math> || With this stipulation, we get the cylinder on <math>X</math>, i.e., the product of <math>X</math> and the [[unit interval]]
+|}
-==Particular cases==
+===The join===
+The [[join]] of two spaces <math>A</math> and <math>B</math> can be constructed as a double mapping cylinder as follow: Set <math>X = A \times B</math>, <math>Y = A</math> and <math>Z = B</math>, and let <math>f,g</math> be the coordinate projection maps.
-* [[Mapping cylinder]]: Here <math>X = Y</math> and <math>f</math> is the identity map
-* [[Mapping cone]]: Here <math>Z</math> is a one-point space and <math>f</math> is the map to that one point
-* [[Join]]: The join of spaces <math>A</math> and <math>B</math> is the double mapping cylinder where <math>X = A \times B</math>, <math>Y = A</math>, <math>Z = B</math> and the maps are simply projections onto the coordinates
-* [[Suspension]]: Here <math>Y</math> and <math>Z</math> are both one-point spaces
 ==Generalizations==
@@ Line 16: / Line 35: @@
 * [[Mapping telescope]]
 ==Facts==
 There is a relation between the homology of the double mapping cylinder of <math>f</math> and <math>g</math>, and the homologies of the spaces <math>X</math>, <math>Y</math> and <math>Z</math>. The relation is described by the [[exact sequence for double mapping cylinder]].