Difference between revisions of "Mapping-cone construction"
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− | + | 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 & 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 & 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 | + | 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 | + | 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 | 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 | 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 | + | 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 | + | 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 |
Mapping-cone construction. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Mapping-cone_construction&oldid=19101