Homotopy fiber

(Redirected from Mapping 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.

Construction

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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   (for  ) a path such that  . We give   a topology by giving it the subspace topology as a subset of   (where   is the space of paths in   which as a function space has the compact-open topology). 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 (which is only well-defined up to homotopy equivalence) 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

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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

 

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

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Homotopy fiber of a fibration

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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 (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.

Duality with mapping cone

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The homotopy fiber is dual to the mapping cone, much as the mapping path space is dual to the mapping cylinder.[3]

Examples

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Loop space

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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

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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

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Postnikov tower

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One main application of the homotopy fiber is in the construction of the Postnikov tower. For a (nice enough) 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

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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  .

See also

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References

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  1. ^ Joseph J. Rotman, An Introduction to Algebraic Topology (1988) Springer-Verlag ISBN 0-387-96678-1 (See Chapter 11 for construction.)
  2. ^ Dugger, Daniel. "A Primer on Homotopy Colimits" (PDF). Archived (PDF) from the original on 3 Dec 2020.
  3. ^ J.P. May, A Concise Course in Algebraic Topology, (1999) Chicago Lectures in Mathematics ISBN 0-226-51183-9 (See chapters 6,7.)