In
mathematics, especially
homotopy theory, the homotopy fiber (sometimes called the mapping fiber)
[Joseph J. Rotman, ''An Introduction to Algebraic Topology'' (1988) Springer-Verlag ''(See Chapter 11 for construction.)''] is part of a construction that associates a
fibration to an arbitrary
continuous function of
topological spaces . It acts as a homotopy theoretic kernel of a mapping of topological spaces due to the fact it yields a long exact sequence of homotopy groups
Moreover, the homotopy fiber can be found in other contexts, such as homological algebra, where the distinguished triangle
gives a long exact sequence analogous to the long exact sequence of homotopy groups. There is a
dual
Dual or Duals may refer to:
Paired/two things
* Dual (mathematics), a notion of paired concepts that mirror one another
** Dual (category theory), a formalization of mathematical duality
*** see more cases in :Duality theories
* Dual (grammatical ...
construction called the
homotopy cofiber.
Construction
The homotopy fiber has a simple description for a continuous map
. If we replace
by a fibration, then the homotopy fiber is simply the fiber of the replacement fibration. We recall this construction of replacing a map by a fibration:
Given such a map, we can replace it with a
fibration by defining the
mapping path space to be the set of pairs
where
and
(for