Homotopy fiber
In mathematics, especially homotopy theory, the homotopy fiber is part of a construction that associates a fibration to an arbitrary continuous function of topological spaces. It is dual to the mapping cone.
In particular, given such a map, define the mapping path space to be the set of pairs where and is a path such that. We give a topology by giving it the subspace topology as a subset of . Then the map given by is a fibration. Furthermore, is homotopy equivalent to as follows: Embed as a subspace of by where is the constant path at. Then deformation retracts to this subspace by contracting the paths.
The fiber of this fibration is the homotopy fiber, which can be defined as the set of all with and a path such that and, where is some fixed basepoint of.
In the special case that the original map was a fibration with fiber, then the homotopy equivalence given above will be a map of fibrations over. This will induce a morphism of their long exact sequences of homotopy groups, from which one can see that the map is a weak equivalence. Thus the above given construction reproduces the same homotopy type if there already is one.
The homotopy fiber is dual to the mapping cone, much as the mapping path space is dual to the mapping cylinder.
