The mapping cone of a continuous map 
is the space formed by gluing the base of the cone
on 
to 
using 
. For based spaces the reduced mapping cone collapses the cone
on the basepoint as well. It models the homotopy cofiber of 
. For an inclusion of a well-behaved pair, it is homotopy equivalent
to the quotient 
.
Mapping cones therefore provide the space-level model for cofiber sequences in stable
homotopy theory.
Mapping Cone
See also
Cofiber, Homotopy, Topological SpectrumExplore with Wolfram|Alpha
References
Hatcher, A. Algebraic Topology. Cambridge, England: Cambridge University Press, 2002.Cite this as:
Weisstein, Eric W. "Mapping Cone." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/MappingCone.html