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Difference between revisions of "Mapping-cone construction"

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The algebraic analogue of the mapping-cone construction is as follows.
 
The algebraic analogue of the mapping-cone construction is as follows.
  
Let  $  u:  K _ {bold \cdot }  \rightarrow L _ {bold \cdot }  $
+
Let  $  u:  K _ {\bullet }  \rightarrow L _ {\bullet }  $
 
be a morphism of complexes, i.e.  $  u = ( u _ {n} ) _ {n \in \mathbf Z }  $
 
be a morphism of complexes, i.e.  $  u = ( u _ {n} ) _ {n \in \mathbf Z }  $
and  $  u _ {n-} 1 \partial  _ {n} = \partial  _ {n} u _ {n} $,  
+
and  $  u _ {n-1} \partial  _ {n} = \partial  _ {n} u _ {n} $,  
where  $  \partial  _ {n}  ^ {K} :  K _ {n} \rightarrow K _ {n-} 1 $.  
+
where  $  \partial  _ {n}  ^ {K} :  K _ {n} \rightarrow K _ {n-1} $.  
 
The mapping cone of  $  u $
 
The mapping cone of  $  u $
is the complex  $  C( u) _ {bold \cdot }  $
+
is the complex  $  C( u) _ {\bullet }  $
 
defined by
 
defined by
  
 
$$  
 
$$  
C( u) _ {n}  =  K _ {n-} 1 \oplus L _ {n} ,\ \  
+
C( u) _ {n}  =  K _ {n-1} \oplus L _ {n} ,\ \  
 
\partial  ( k, l)  =  ( - \partial  k , \partial  l + uk ).
 
\partial  ( k, l)  =  ( - \partial  k , \partial  l + uk ).
 
$$
 
$$
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The injections  $  L _ {n} \rightarrow C( u) _ {n} $
 
The injections  $  L _ {n} \rightarrow C( u) _ {n} $
 
define a morphism of complexes and if  $  K [ - 1 ] $
 
define a morphism of complexes and if  $  K [ - 1 ] $
denotes the complex with  $  K [ - 1 ] _ {n} = K _ {n-} 1 $
+
denotes the complex with  $  K [ - 1 ] _ {n} = K _ {n-1} $
and  $  \partial  _ {n} ^ {K[ - 1 ] } = - \partial  _ {n-} 1 ^ {K} $,  
+
and  $  \partial  _ {n} ^ {K[ - 1 ] } = - \partial  _ {n-1}  ^ {K} $,  
then the projections  $  C( u) _ {n} \rightarrow K _ {n-} 1 $
+
then the projections  $  C( u) _ {n} \rightarrow K _ {n-1} $
 
yield
 
yield
  
Line 74: Line 74:
  
 
$$  
 
$$  
0  \rightarrow  L _ {bold \cdot }  \rightarrow ^ { i }  C( u) _ {bold \cdot }  \rightarrow ^ { p }  K[ - 1] _ {bold \cdot }  \rightarrow  0,
+
0  \rightarrow  L _ {\bullet }  \rightarrow ^ { i }  C( u) _ {\bullet }  \rightarrow ^ { p }  K[ - 1] _ {\bullet }  \rightarrow  0,
 
$$
 
$$
  
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$$  
 
$$  
 
\dots \rightarrow \  
 
\dots \rightarrow \  
H _ {n} ( L _ { bold \cdot } )  \rightarrow ^ { {i _ * } } \  
+
H _ {n} ( L _ { \bullet } )  \rightarrow ^ { {i _ * } } \  
H _ {n} ( C( u) _ { bold \cdot } )  \rightarrow ^ { {p _ * } } \  
+
H _ {n} ( C( u) _ { \bullet } )  \rightarrow ^ { {p _ * } } \  
H _ {n-} 1 ( K _ { bold \cdot } )  \rightarrow ^ { {u _ * } } \  
+
H _ {n-1} ( K _ { \bullet } )  \rightarrow ^ { {u _ * } } \  
H _ {n-} 1 ( L _ {bold \cdot }  )  \rightarrow \dots .
+
H _ {n-1} ( L _ {\bullet }  )  \rightarrow \dots .
 
$$
 
$$
  
By turning a complex  $  K _ {bold \cdot }  $
+
By turning a complex  $  K _ {\bullet }  $
into a  "co-complex"  $  K ^ { bold \cdot } $,  
+
into a  "co-complex"  $  K ^ { \bullet } $,  
$  K  ^ {n} = K _ {-} n $,  
+
$  K  ^ {n} = K _ {-n} $,  
 
the analogous constructions and results in a cohomological setting are obtained.
 
the analogous constructions and results in a cohomological setting are obtained.
  
The complex  $  K [ - 1] _ {bold \cdot }  $
+
The complex  $  K [ - 1] _ {\bullet }  $
is called the suspension of the complex  $  K _ {bold \cdot }  $.
+
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=51293
This article was adapted from an original article by A.F. Kharshiladze (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article