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The construction that associates with every continuous mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062280/m0622801.png" /> of topological spaces the topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062280/m0622802.png" /> obtained from the topological sum (disjoint union) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062280/m0622803.png" /> (here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062280/m0622804.png" /> is the [[Cone|cone]] over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062280/m0622805.png" />) by identifying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062280/m0622806.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062280/m0622807.png" />. The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062280/m0622808.png" /> is called the mapping cone of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062280/m0622809.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062280/m06228010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062280/m06228011.png" /> are pointed spaces with distinguished points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062280/m06228012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062280/m06228013.png" />, then the generator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062280/m06228014.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062280/m06228015.png" /> is contracted to a point, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062280/m06228016.png" /> is said to be the collapsed mapping cone of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062280/m06228017.png" />. For an arbitrary pointed topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062280/m06228018.png" />, the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062280/m06228019.png" /> induces an exact sequence
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062280/m06228020.png" /></td> </tr></table>
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of pointed sets. The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062280/m06228021.png" /> is homotopic to the constant mapping to the distinguished point if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062280/m06228022.png" /> is a retract of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062280/m06228023.png" /> (cf. [[Retract of a topological space|Retract of a topological space]]).
+
The construction that associates with every continuous mapping $  f :  X \rightarrow Y $
 +
of topological spaces the topological space  $  C _ {f} \supset Y $
 +
obtained from the topological sum (disjoint union)  $  C X \oplus Y $(
 +
here  $  C X = ( X \times [ 0 , 1 ] ) / ( X \times \{ 0 \} ) $
 +
is the [[Cone|cone]] over  $  X $)
 +
by identifying  $  x \times \{ 1 \} = f ( x) $,
 +
$  x \in X $.
 +
The space  $  C _ {f} $
 +
is called the mapping cone of  $  f $.  
 +
If  $  X $
 +
and  $  Y $
 +
are pointed spaces with distinguished points  $  x \in X $,
 +
$  y \in Y $,
 +
then the generator  $  x \times [ 0 , 1 ] $
 +
of  $  C X $
 +
is contracted to a point, and  $  C _ {f} $
 +
is said to be the collapsed mapping cone of  $  f $.  
 +
For an arbitrary pointed topological space  $  K $,
 +
the sequence  $  X \rightarrow  ^ {f} Y \subset  C _ {f} $
 +
induces an exact sequence
 +
 
 +
$$
 +
[ X , K ]  \leftarrow  [ Y , K ]  \leftarrow  [ C _ {f} , K ]
 +
$$
 +
 
 +
of pointed sets. The mapping  $  f $
 +
is homotopic to the constant mapping to the distinguished point if and only if $  Y $
 +
is a retract of $  C _ {f} $(
 +
cf. [[Retract of a topological space|Retract of a topological space]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E.H. Spanier,  "Algebraic topology" , McGraw-Hill  (1966)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M.K. Tangora,  "Cohomology operations and their applications in homotopy theory" , Harper &amp; Row  (1968)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E.H. Spanier,  "Algebraic topology" , McGraw-Hill  (1966)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M.K. Tangora,  "Cohomology operations and their applications in homotopy theory" , Harper &amp; Row  (1968)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
The algebraic analogue of the mapping-cone construction is as follows.
 
The algebraic analogue of the mapping-cone construction is as follows.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062280/m06228024.png" /> be a morphism of complexes, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062280/m06228025.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062280/m06228026.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062280/m06228027.png" />. The mapping cone of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062280/m06228028.png" /> is the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062280/m06228029.png" /> defined by
+
Let $  u: K _ {\bullet }  \rightarrow L _ {\bullet }  $
 +
be a morphism of complexes, i.e. $  u = ( u _ {n} ) _ {n \in \mathbf Z }  $
 +
and $  u _ {n-1} \partial  _ {n} = \partial  _ {n} u _ {n} $,  
 +
where $  \partial  _ {n}  ^ {K} : K _ {n} \rightarrow K _ {n-1} $.  
 +
The mapping cone of $  u $
 +
is the complex $  C( u) _ {\bullet }  $
 +
defined by
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062280/m06228030.png" /></td> </tr></table>
+
$$
 +
C( u) _ {n}  = K _ {n-1} \oplus L _ {n} ,\ \
 +
\partial  ( k, l)  = ( - \partial  k , \partial  l + uk ).
 +
$$
  
The injections <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062280/m06228031.png" /> define a morphism of complexes and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062280/m06228032.png" /> denotes the complex with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062280/m06228033.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062280/m06228034.png" />, then the projections <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062280/m06228035.png" /> yield
+
The injections $  L _ {n} \rightarrow C( u) _ {n} $
 +
define a morphism of complexes and if $  K [ - 1 ] $
 +
denotes the complex with $  K [ - 1 ] _ {n} = K _ {n-1} $
 +
and $  \partial  _ {n} ^ {K[ - 1 ] } = - \partial  _ {n-1}  ^ {K} $,  
 +
then the projections $  C( u) _ {n} \rightarrow K _ {n-1} $
 +
yield
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062280/m06228036.png" /></td> </tr></table>
+
$$
 +
0  \rightarrow  L _ {n}  \rightarrow  C( u) _ {n}  \rightarrow  K[ - 1] _ {n}  \rightarrow  0,
 +
$$
  
 
which fit together to define a short exact sequence of complexes
 
which fit together to define a short exact sequence of complexes
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062280/m06228037.png" /></td> </tr></table>
+
$$
 +
0  \rightarrow  L _ {\bullet }  \rightarrow ^ { i }  C( u) _ {\bullet }  \rightarrow ^ { p }  K[ - 1] _ {\bullet }  \rightarrow  0,
 +
$$
  
 
and there results a long exact homology sequence
 
and there results a long exact homology sequence
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062280/m06228038.png" /></td> </tr></table>
+
$$
 +
\dots \rightarrow \
 +
H _ {n} ( L _ { \bullet } )  \rightarrow ^ { {i _ * } } \
 +
H _ {n} ( C( u) _ { \bullet } )  \rightarrow ^ { {p _ * } } \
 +
H _ {n-1} ( K _ { \bullet } )  \rightarrow ^ { {u _ * } } \
 +
H _ {n-1} ( L _ {\bullet }  )  \rightarrow \dots .
 +
$$
  
By turning a complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062280/m06228039.png" /> into a  "co-complex"  <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062280/m06228040.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062280/m06228041.png" />, the analogous constructions and results in a cohomological setting are obtained.
+
By turning a complex $  K _ {\bullet }  $
 +
into a  "co-complex"   $ K ^ { \bullet } $,  
 +
$  K  ^ {n} = K _ {-n} $,  
 +
the analogous constructions and results in a cohomological setting are obtained.
  
The complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062280/m06228042.png" /> is called the suspension of the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062280/m06228043.png" />.
+
The complex $  K [ - 1] _ {\bullet }  $
 +
is called the suspension of the complex $  K _ {\bullet }  $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. MacLane,  "Homology" , Springer  (1963)  pp. Sect. II.4</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. MacLane,  "Homology" , Springer  (1963)  pp. Sect. II.4</TD></TR></table>

Latest revision as of 12:17, 12 January 2021


The construction that associates with every continuous mapping $ f : X \rightarrow Y $ of topological spaces the topological space $ C _ {f} \supset Y $ obtained from the topological sum (disjoint union) $ C X \oplus Y $( here $ C X = ( X \times [ 0 , 1 ] ) / ( X \times \{ 0 \} ) $ is the cone over $ X $) by identifying $ x \times \{ 1 \} = f ( x) $, $ x \in X $. The space $ C _ {f} $ is called the mapping cone of $ f $. If $ X $ and $ Y $ are pointed spaces with distinguished points $ x \in X $, $ y \in Y $, then the generator $ x \times [ 0 , 1 ] $ of $ C X $ is contracted to a point, and $ C _ {f} $ is said to be the collapsed mapping cone of $ f $. For an arbitrary pointed topological space $ K $, the sequence $ X \rightarrow ^ {f} Y \subset C _ {f} $ induces an exact sequence

$$ [ X , K ] \leftarrow [ Y , K ] \leftarrow [ C _ {f} , K ] $$

of pointed sets. The mapping $ f $ is homotopic to the constant mapping to the distinguished point if and only if $ Y $ is a retract of $ C _ {f} $( cf. Retract of a topological space).

References

[1] E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966)
[2] M.K. Tangora, "Cohomology operations and their applications in homotopy theory" , Harper & Row (1968)

Comments

The algebraic analogue of the mapping-cone construction is as follows.

Let $ u: K _ {\bullet } \rightarrow L _ {\bullet } $ be a morphism of complexes, i.e. $ u = ( u _ {n} ) _ {n \in \mathbf Z } $ and $ u _ {n-1} \partial _ {n} = \partial _ {n} u _ {n} $, where $ \partial _ {n} ^ {K} : K _ {n} \rightarrow K _ {n-1} $. The mapping cone of $ u $ is the complex $ C( u) _ {\bullet } $ defined by

$$ C( u) _ {n} = K _ {n-1} \oplus L _ {n} ,\ \ \partial ( k, l) = ( - \partial k , \partial l + uk ). $$

The injections $ L _ {n} \rightarrow C( u) _ {n} $ define a morphism of complexes and if $ K [ - 1 ] $ denotes the complex with $ K [ - 1 ] _ {n} = K _ {n-1} $ and $ \partial _ {n} ^ {K[ - 1 ] } = - \partial _ {n-1} ^ {K} $, then the projections $ C( u) _ {n} \rightarrow K _ {n-1} $ yield

$$ 0 \rightarrow L _ {n} \rightarrow C( u) _ {n} \rightarrow K[ - 1] _ {n} \rightarrow 0, $$

which fit together to define a short exact sequence of complexes

$$ 0 \rightarrow L _ {\bullet } \rightarrow ^ { i } C( u) _ {\bullet } \rightarrow ^ { p } K[ - 1] _ {\bullet } \rightarrow 0, $$

and there results a long exact homology sequence

$$ \dots \rightarrow \ H _ {n} ( L _ { \bullet } ) \rightarrow ^ { {i _ * } } \ H _ {n} ( C( u) _ { \bullet } ) \rightarrow ^ { {p _ * } } \ H _ {n-1} ( K _ { \bullet } ) \rightarrow ^ { {u _ * } } \ H _ {n-1} ( L _ {\bullet } ) \rightarrow \dots . $$

By turning a complex $ K _ {\bullet } $ into a "co-complex" $ K ^ { \bullet } $, $ K ^ {n} = K _ {-n} $, the analogous constructions and results in a cohomological setting are obtained.

The complex $ K [ - 1] _ {\bullet } $ is called the suspension of the complex $ K _ {\bullet } $.

References

[a1] S. MacLane, "Homology" , Springer (1963) pp. Sect. II.4
How to Cite This Entry:
Mapping-cone construction. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Mapping-cone_construction&oldid=19101
This article was adapted from an original article by A.F. Kharshiladze (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article