# 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)

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