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).
|||E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966)|
|||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 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 .
|[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