Homotopy fiber

In mathematics, especially homotopy theory, the homotopy fiber (sometimes called the mapping fiber) is part of a construction that associates a fibration to an arbitrary continuous function of topological spaces formula_1. 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 groupsformula_2Moreover, the homotopy fiber can be found in other contexts, such as homological algebra, where the distinguished triangleformula_3gives 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 formula_1. If we replace formula_5 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 formula_6 to be the set of pairs formula_7 where formula_8 and formula_9 (for formula_10) a path such that formula_11. We give formula_6 a topology by giving it the subspace topology as a subset of formula_13 (where formula_14 is the space of paths in formula_15 which as a function space has the compact-open topology). Then the map formula_16 given by formula_17 is a fibration. Furthermore, formula_6 is homotopy equivalent to formula_19 as follows: Embed formula_19 as a subspace of formula_6 by formula_22 where formula_23 is the constant path at formula_24. Then formula_6 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 fiberformula_26which can be defined as the set of all formula_7 with formula_8 and formula_9 a path such that formula_11 and formula_31 for some fixed basepoint formula_32.
As a homotopy limit.
Another way to construct the homotopy fiber of a map is to consider the homotopy limitpg 21 of the diagramformula_33this is because computing the homotopy limit amounts to finding the pullback of the diagramformula_34where the vertical map is the source and target map of a path formula_35, soformula_36This means the homotopy limit is in the collection of mapsformula_37which is exactly the homotopy fiber as defined above.
Properties.
Homotopy fiber of a fibration.
In the special case that the original map formula_5 was a fibration with fiber formula_39, then the homotopy equivalence formula_40 given above will be a map of fibrations over formula_15. 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 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 formula_42 and the inclusion of a pointformula_43the homotopy fiber of this map is thenformula_44which is the loop space formula_45.
From a covering space.
Given a universal coveringformula_46the homotopy fiber formula_47 has the propertyformula_48which 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 (nice enough) topological space formula_42, we can construct a sequence of spaces formula_50 and maps formula_51 whereformula_52andformula_53Now, these maps formula_54 can be iteratively constructed using homotopy fibers. This is because we can take a mapformula_55representing a cohomology class informula_56and construct the homotopy fiberformula_57In addition, notice the homotopy fiber of formula_51 isformula_59showing 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 formula_60 and maps formula_61 whereformula_62hence formula_63. If we take the induced mapformula_64the homotopy fiber of this map recovers the formula_65-th postnikov approximation formula_66 since the long exact sequence of the fibrationformula_67we get
formula_68
which gives isomorphismsformula_69for formula_70.