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. 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.

Construction

The homotopy fiber has a simple description for a continuous map. If we replace 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 to be the set of pairs where and a path such that. We give a topology by giving it the subspace topology as a subset of . Then the map given by is a fibration. Furthermore, is homotopy equivalent to as follows: Embed as a subspace of by where is the constant path at. Then deformation retracts to this subspace by contracting the paths.
The fiber of this fibration is the homotopy fiber

which can be defined as the set of all with and a path such that and for some fixed basepoint. A consequence of this definition is that if two points of are in the same path connected component, then their homotopy fibers are homotopy equivalent.

As a homotopy limit

Another way to construct the homotopy fiber of a map is to consider the homotopy limit^{pg 21} of the diagram

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

where the vertical map is the source and target map of a path, so

This means the homotopy limit is in the collection of maps

which is exactly the homotopy fiber as defined above.
If and can be connected by a path in, then the diagrams

and

are homotopy equivalent to the diagram

and thus the homotopy fibers of and are isomorphic in. Therefore we often speak about the homotopy fiber of a map without specifying a base point.

Properties

Homotopy fiber of a fibration

In the special case that the original map was a fibration with fiber, then the homotopy equivalence given above will be a map of fibrations over. This will induce a morphism of their long exact sequences of homotopy groups, from which one can see that the map is a weak equivalence. Thus the above given construction reproduces the same homotopy type if there already is one.

Duality with mapping cone

The homotopy fiber is dual to the mapping cone, much as the mapping path space is dual to the mapping cylinder.

Examples

Loop space

Given a topological space and the inclusion of a point

the homotopy fiber of this map is then

which is the loop space.
See also: Path space fibration.

From a covering space

Given a universal covering

the homotopy fiber has the property

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.

Applications

Postnikov tower

One main application of the homotopy fiber is in the construction of the Postnikov tower. For a topological space, we can construct a sequence of spaces and maps where

and

Now, these maps can be iteratively constructed using homotopy fibers. This is because we can take a map

representing a cohomology class in

and construct the homotopy fiber

In addition, notice the homotopy fiber of is

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.

Maps from the whitehead tower

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

hence. If we take the induced map

the homotopy fiber of this map recovers the -th postnikov approximation since the long exact sequence of the fibration

we get

which gives isomorphisms

for.