Homotopy fiber

In mathematics, especially homotopy theory, the homotopy fiber (sometimes called the mapping fiber)^[1] is part of a construction that associates a fibration to an arbitrary continuous function of topological spaces f : A → B. It is dual to the mapping cone.

In particular, given such a map, define the mapping path space E_f to be the set of pairs (a, p) where a ∈ A and p : [0,1] → B is a path such that p(0) = f(a). We give E_f a topology by giving it the subspace topology as a subset of A × B^I (where B^I is the space of paths in B which as a function space has the compact-open topology). Then the map E_f → B given by (a, p) ⟼ p(1) is a fibration. Furthermore, E_f is homotopy equivalent to A as follows: Embed A as a subspace of E_f by a ⟼ (a, p_a) where p_a is the constant path at f(a). Then E_f deformation retracts to this subspace by contracting the paths.

The fiber of this fibration (which is only well-defined up to homotopy equivalence) is the homotopy fiber F_f, which can be defined as the set of all (a, p) with a ∈ A and p : [0,1] → B a path such that p(0) = f(a) and p(1) = b₀, where b₀ ∈ B is some fixed basepoint of B.

In the special case that the original map f was a fibration with fiber F, then the homotopy equivalence A → E_f given above will be a map of fibrations over B. This will induce a morphism of their long exact sequences of homotopy groups, from which (by applying the Five Lemma, as is done in the Puppe sequence) one can see that the map F → F_f 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.^[2]