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 f : A → B.

In particular, given such a map, define Ef 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 Ef a topology by giving it the subspace topology as a subset of A × BI (where BI is the space of paths in B which as a function space has the compact-open topology). Then the map Ef → B given by (a, p) ⟼ p(1) is a fibration. Furthermore, Ef is homotopy equivalent to A as follows: Embed A as a subspace of Ef by a ⟼ (a, pa) where pa is the constant path at f(a). Then Ef 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 Ff, 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 → Ef 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) one can see that the map F → Ff is a weak equivalence. Thus the above given construction reproduces the same homotopy type if there already is one.

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