A homotopy cofiber of a based map 
is the homotopy pushout of 
. For spaces it can be modeled by the mapping
cone, obtained by attaching the cone on 
to 
using 
. For spectra,
a cofiber sequence 
extends to 
and induces a long exact sequence of homotopy
groups. The cofiber of multiplication by 
on the sphere spectrum
is the mod-
Moore spectrum. Successive cofibers specify
the pieces of a filtration, while the attaching maps record whether those pieces
split.
Cofiber
See also
Mapping Cone, Sphere Spectrum, Thick Subcategory, Topological SpectrumExplore with Wolfram|Alpha
References
May, J. P. A Concise Course in Algebraic Topology. Chicago, IL: University of Chicago Press, 1999.Cite this as:
Weisstein, Eric W. "Cofiber." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Cofiber.html