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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 : AB. It is dual to the mapping cone.

In particular, given such a map, define the mapping path space Ef to be the set of pairs (a, p) where aA 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 EfB 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 aA and p : [0,1] → B a path such that p(0) = f(a) and p(1) = b0, where b0B 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 AEf 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 FFf 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]

Quasi-fibration

References

Joseph J. Rotman, An Introduction to Algebraic Topology (1988) Springer-Verlag ISBN 0-387-96678-1 (See Chapter 11 for construction.)

J.P. May, A Concise Course in Algebraic Topology, (1999) Chicago Lectures in Mathematics ISBN 0-226-51183-9 (See chapters 6,7.)

Hatcher, Allen (2002), Algebraic Topology, Cambridge: Cambridge University Press, ISBN 0-521-79540-0.