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 \colon A \to B.

In particular, given such a map, define E_f to be the set of pairs (a,p) where a \in A and p \colon [0,1] \to 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 \times 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 \to B given by (a,p) \mapsto 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 \mapsto (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 \in A and p \colon [0,1] \to B a path such that p(0) = f(a) and p(1) = b_0, where b_0 \in B is some fixed basepoint of B.

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