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

It acts as a homotopy theoretic kernel of a mapping of topological spaces due to the fact it yields a long exact sequence of homotopy groups

Moreover, the homotopy fiber can be found in other contexts, such as homological algebra, where the distinguished triangle

gives a long exact sequence analogous to the long exact sequence of homotopy groups.

There is a dual construction called the homotopy cofiber.

The homotopy fiber has a simple description for a continuous map

by a fibration, then the homotopy fiber is simply the fiber of the replacement fibration.

We recall this construction of replacing a map by a fibration: Given such a map, we can replace it with a fibration by defining the mapping path space

which as a function space has the compact-open topology).

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

Another way to construct the homotopy fiber of a map is to consider the homotopy limit[2]pg 21 of the diagram

this is because computing the homotopy limit amounts to finding the pullback of the diagram

This means the homotopy limit is in the collection of maps

which is exactly the homotopy fiber as defined above.

Therefore we often speak about the homotopy fiber of a map without specifying a base point.

In the special case that the original map

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 → Ff is a weak equivalence.

Thus the above given construction reproduces the same homotopy type if there already is one.

which can be seen by looking at the long exact sequence of the homotopy groups for the fibration.

This is analyzed further below by looking at the Whitehead tower.

One main application of the homotopy fiber is in the construction of the Postnikov tower.

For a (nice enough) topological space

can be iteratively constructed using homotopy fibers.

In addition, notice the homotopy fiber of

showing the homotopy fiber acts like a homotopy-theoretic kernel.

Note this fact can be shown by looking at the long exact sequence for the fibration constructing the homotopy fiber.

The dual notion of the Postnikov tower is the Whitehead tower which gives a sequence of spaces

the homotopy fiber of this map recovers the

since the long exact sequence of the fibration