Mapping-cone construction

The construction that associates with every continuous mapping of topological spaces the topological space obtained from the topological sum (disjoint union) (here is the cone over ) by identifying , . The space is called the mapping cone of . If and are pointed spaces with distinguished points , , then the generator of is contracted to a point, and is said to be the collapsed mapping cone of . For an arbitrary pointed topological space , the sequence induces an exact sequence

of pointed sets. The mapping is homotopic to the constant mapping to the distinguished point if and only if is a retract of (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 be a morphism of complexes, i.e. and , where . The mapping cone of is the complex defined by

The injections define a morphism of complexes and if denotes the complex with and , then the projections yield

which fit together to define a short exact sequence of complexes

and there results a long exact homology sequence

By turning a complex into a "co-complex" , , the analogous constructions and results in a cohomological setting are obtained.

The complex is called the suspension of the complex .

References

[a1]	S. MacLane, "Homology" , Springer (1963) pp. Sect. II.4