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
![{\displaystyle \cdots \to \pi _{n+1}(B)\to \pi _{n}({\text{Hofiber}}(f))\to \pi _{n}(A)\to \pi _{n}(B)\to \cdots }](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/045d83a7b24c2dafab4c66e9b33f78f13977a258)
Moreover, the homotopy fiber can be found in other contexts, such as homological algebra, where the distinguished triangle
![{\displaystyle C(f)_{\bullet }[-1]\to A_{\bullet }\to B_{\bullet }\xrightarrow {[+1]} }](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/07ade1436c3800568e006d8d5b4e5dfb0dcdeca3)
gives a long exact sequence analogous to the long exact sequence of homotopy groups. There is a dual construction called the homotopy cofiber.
Construction[edit]
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
![{\displaystyle {\begin{matrix}{\text{Hofiber}}(f)&\to &E_{f}\\&&\downarrow \\&&B\end{matrix}}}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/95234111f1b1f6675dd563002a9b82f094e5c6f8)
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[edit]
Another way to construct the homotopy fiber of a map is to consider the homotopy limit[2]pg 21 of the diagram
![{\displaystyle {\underset {\leftarrow }{\text{holim}}}\left({\begin{matrix}&&*\\&&\downarrow \\A&\xrightarrow {f} &B\end{matrix}}\right)\simeq F_{f}}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/54a3f829906a2619876bff0ad8cd4704c485ced0)
this is because computing the homotopy limit amounts to finding the pullback of the diagram
![{\displaystyle {\begin{matrix}&&B^{I}\\&&\downarrow \\A\times *&\xrightarrow {f} &B\times B\end{matrix}}}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/37f94c09f8dbc64401498769c1df17927415a75b)
where the vertical map is the source and target map of a path
, so
![{\displaystyle \gamma \mapsto (\gamma (0),\gamma (1))}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/bb2d858f4f7bda5c7c230ac22af27116ebfe46c6)
This means the homotopy limit is in the collection of maps
![{\displaystyle \left\{(a,\gamma )\in A\times B^{I}:f(a)=\gamma (0){\text{ and }}\gamma (1)=*\right\}}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/a08f828613219b5d59ec48a484eaf0cd1dfaf041)
which is exactly the homotopy fiber as defined above.
If
and
can be connected by a path
in
, then the diagrams
![{\displaystyle {\begin{matrix}&&x_{0}\\&&\downarrow \\A&\xrightarrow {f} &B\end{matrix}}}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/ea0fcd6da0a3913ff496b8e5b9382877bc43aa85)
and
![{\displaystyle {\begin{matrix}&&x_{1}\\&&\downarrow \\A&\xrightarrow {f} &B\end{matrix}}}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/07cfd91790f14a296ca2bd8f77186ec21cc4fcc8)
are homotopy equivalent to the diagram
![{\displaystyle {\begin{matrix}&&[0,1]\\&&\downarrow {\delta }\\A&\xrightarrow {f} &B\end{matrix}}}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/50589073ffe77f310604e4b2e072bf2def3a89e4)
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[edit]
Homotopy fiber of a fibration[edit]
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 F → Ff is a weak equivalence. Thus the above given construction reproduces the same homotopy type if there already is one.
Duality with mapping cone[edit]
The homotopy fiber is dual to the mapping cone, much as the mapping path space is dual to the mapping cylinder.[3]
Examples[edit]
Loop space[edit]
Given a topological space
and the inclusion of a point
![{\displaystyle \iota :\{x_{0}\}\hookrightarrow X}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/d558aeed39684537f70e41fdacad2a96dcd41b04)
the homotopy fiber of this map is then
![{\displaystyle \left\{(x_{0},\gamma )\in \{x_{0}\}\times X^{I}:x_{0}=\gamma (0){\text{ and }}\gamma (1)=x_{0}\right\}}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/58a8046f938925cbdf749f3d63a53519a5af0cbc)
which is the loop space
.
From a covering space[edit]
Given a universal covering
![{\displaystyle \pi :{\tilde {X}}\to X}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/d5acd2230dd5763a94d37a7da82e513539ae3e34)
the homotopy fiber
has the property
![{\displaystyle \pi _{k}({\text{Hofiber}}(\pi ))={\begin{cases}\pi _{0}(X)&k<1\\0&k\geq 1\end{cases}}}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/209170e027edfbcdcdef9b79910a936356b2306f)
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[edit]
Postnikov tower[edit]
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
![{\displaystyle \pi _{k}\left(X_{n}\right)={\begin{cases}\pi _{k}(X)&k\leq n\\0&{\text{ otherwise }}\end{cases}}}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/5e8574c6dd2481895798ab96ae2cd8258653de54)
and
![{\displaystyle X\simeq {\underset {\leftarrow }{\text{lim}}}\left(X_{k}\right)}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/0d5f8f909ae0423855611f2ec004897337f10e1a)
Now, these maps
can be iteratively constructed using homotopy fibers. This is because we can take a map
![{\displaystyle X_{n-1}\to K\left(\pi _{n}(X),n-1\right)}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/c95a82178a562e7ef7a0d09a1cbe260f50c2b8c7)
representing a cohomology class in
![{\displaystyle H^{n-1}\left(X_{n-1},\pi _{n}(X)\right)}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/74c2a617ef8282eed5d13dfdaee181aa73a75b2b)
and construct the homotopy fiber
![{\displaystyle {\underset {\leftarrow }{\text{holim}}}\left({\begin{matrix}&&*\\&&\downarrow \\X_{n-1}&\xrightarrow {f} &K\left(\pi _{n}(X),n-1\right)\end{matrix}}\right)\simeq X_{n}}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/aad4642beecde58bf4a49a2a0a0ce096c06e0da9)
In addition, notice the homotopy fiber of
is
![{\displaystyle {\text{Hofiber}}\left(f_{n}\right)\simeq K\left(\pi _{n}(X),n\right)}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/5f1e67cfa470fcb394f93480e09286ca347ee793)
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[edit]
The dual notion of the Postnikov tower is the Whitehead tower which gives a sequence of spaces
and maps
where
![{\displaystyle \pi _{k}\left(X^{n}\right)={\begin{cases}\pi _{k}(X)&k\geq n\\0&{\text{otherwise}}\end{cases}}}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/ee24cda5d768c836f8c2099d71a7f8d4716635ac)
hence
. If we take the induced map
![{\displaystyle f_{0}^{n+1}:X^{n+1}\to X}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/1a831f9babf6aa7485e77b412457ecdb0fe74262)
the homotopy fiber of this map recovers the
-th postnikov approximation
since the long exact sequence of the fibration
![{\displaystyle {\begin{matrix}{\text{Hofiber}}\left(f_{0}^{n+1}\right)&\to &X^{n+1}\\&&\downarrow \\&&X\end{matrix}}}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/e41c220877ffa9bd165f4e74d299722a6db81a94)
we get
![{\displaystyle {\begin{matrix}\to &\pi _{k+1}\left({\text{Hofiber}}\left(f_{0}^{n+1}\right)\right)&\to &\pi _{k+1}(X^{n+1})&\to &\pi _{k+1}(X)&\to \\&\pi _{k}\left({\text{Hofiber}}\left(f_{0}^{n+1}\right)\right)&\to &\pi _{k}\left(X^{n+1}\right)&\to &\pi _{k}(X)&\to \\&\pi _{k-1}\left({\text{Hofiber}}\left(f_{0}^{n+1}\right)\right)&\to &\pi _{k-1}\left(X^{n+1}\right)&\to &\pi _{k-1}(X)&\to \end{matrix}}}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/dd481f0ea79021eeb4faf44767c380b1877e609f)
which gives isomorphisms
![{\displaystyle \pi _{k-1}\left({\text{Hofiber}}\left(f_{0}^{n+1}\right)\right)\cong \pi _{k}(X)}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/3c0e7479364d510c79403400ed549c1c82f3ca75)
for
.
See also[edit]
References[edit]