Mapping-cone construction

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