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 ${\displaystyle f:A\to B}$. 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

${\displaystyle \cdots \to \pi _{n+1}(B)\to \pi _{n}({\text{Hofiber}}(f))\to \pi _{n}(A)\to \pi _{n}(B)\to \cdots }$

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

${\displaystyle C(f)_{\bullet }[-1]\to A_{\bullet }\to B_{\bullet }\xrightarrow {[+1]} }$

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 ${\displaystyle f:A\to B}$. If we replace ${\displaystyle f}$ 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 ${\displaystyle E_{f}}$ to be the set of pairs ${\displaystyle (a,\gamma )}$ where ${\displaystyle a\in A}$ and ${\displaystyle \gamma :I\to B}$ (for ${\displaystyle I=[0,1]}$) a path such that ${\displaystyle \gamma (0)=f(a)}$. We give ${\displaystyle E_{f}}$ a topology by giving it the subspace topology as a subset of ${\displaystyle A\times B^{I}}$ (where ${\displaystyle B^{I}}$ is the space of paths in ${\displaystyle B}$ which as a function space has the compact-open topology). Then the map ${\displaystyle E_{f}\to B}$ given by ${\displaystyle (a,\gamma )\mapsto \gamma (1)}$ is a fibration. Furthermore, ${\displaystyle E_{f}}$ is homotopy equivalent to ${\displaystyle A}$ as follows: Embed ${\displaystyle A}$ as a subspace of ${\displaystyle E_{f}}$ by ${\displaystyle a\mapsto \gamma _{a}}$ where ${\displaystyle \gamma _{a}}$ is the constant path at ${\displaystyle f(a)}$. Then ${\displaystyle E_{f}}$ 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

${\displaystyle {\begin{matrix}{\text{Hofiber}}(f)&\to &E_{f}\\&&\downarrow \\&&B\end{matrix}}}$

which can be defined as the set of all ${\displaystyle (a,\gamma )}$ with ${\displaystyle a\in A}$ and ${\displaystyle \gamma :I\to B}$ a path such that ${\displaystyle \gamma (0)=f(a)}$ and ${\displaystyle \gamma (1)=*}$ for some fixed basepoint ${\displaystyle *\in B}$.

As a homotopy limit[]

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

${\displaystyle {\underset {\leftarrow }{\text{holim}}}\left({\begin{matrix}&&*\\&&\downarrow \\A&\xrightarrow {f} &B\end{matrix}}\right)\simeq F_{f}}$

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

${\displaystyle {\begin{matrix}&&B^{I}\\&&\downarrow \\A\times *&\xrightarrow {f} &B\times B\end{matrix}}}$

where the vertical map is the source and target map of a path ${\displaystyle \gamma :I\to B}$, so

${\displaystyle \gamma \mapsto (\gamma (0),\gamma (1))}$

This means the homotopy limit is in the collection of maps

${\displaystyle \left\{(a,\gamma )\in A\times B^{I}:f(a)=\gamma (0){\text{ and }}\gamma (1)=*\right\}}$

which is exactly the homotopy fiber as defined above.

Properties[]

Homotopy fiber of a fibration[]

In the special case that the original map ${\displaystyle f}$ was a fibration with fiber ${\displaystyle F}$, then the homotopy equivalence ${\displaystyle A\to E_{f}}$ given above will be a map of fibrations over ${\displaystyle B}$. 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 → F_f 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.^[3]

Examples[]

Loop space[]

Given a topological space ${\displaystyle X}$ and the inclusion of a point

${\displaystyle \iota :\{x_{0}\}\hookrightarrow X}$

the homotopy fiber of this map is then

${\displaystyle \left\{(x_{0},\gamma )\in \{x_{0}\}\times X^{I}:x_{0}=\gamma (0){\text{ and }}\gamma (1)=x_{0}\right\}}$

which is the loop space ${\displaystyle \Omega X}$.

From a covering space[]

Given a universal covering

${\displaystyle \pi :{\tilde {X}}\to X}$

the homotopy fiber ${\displaystyle {\text{Hofiber}}(\pi )}$ has the property

${\displaystyle \pi _{k}({\text{Hofiber}}(\pi ))={\begin{cases}\pi _{0}(X)&k<1\\0&k\geq 1\end{cases}}}$

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 (nice enough) topological space ${\displaystyle X}$, we can construct a sequence of spaces ${\displaystyle \left\{X_{n}\right\}_{n\geq 0}}$ and maps ${\displaystyle f_{n}:X_{n}\to X_{n-1}}$ where

${\displaystyle \pi _{k}\left(X_{n}\right)={\begin{cases}\pi _{k}(X)&k\leq n\\0&{\text{ otherwise }}\end{cases}}}$

and

${\displaystyle X\simeq {\underset {\leftarrow }{\text{lim}}}\left(X_{k}\right)}$

Now, these maps ${\displaystyle f_{n}}$ can be iteratively constructed using homotopy fibers. This is because we can take a map

${\displaystyle X_{n-1}\to K\left(\pi _{n}(X),n-1\right)}$

representing a cohomology class in

${\displaystyle H^{n-1}\left(X_{n-1},\pi _{n}(X)\right)}$

and construct the homotopy fiber

${\displaystyle {\underset {\leftarrow }{\text{holim}}}\left({\begin{matrix}&&*\\&&\downarrow \\X_{n-1}&\xrightarrow {f} &K\left(\pi _{n}(X),n-1\right)\end{matrix}}\right)\simeq X_{n}}$

In addition, notice the homotopy fiber of ${\displaystyle f_{n}:X_{n}\to X_{n-1}}$ is

${\displaystyle {\text{Hofiber}}\left(f_{n}\right)\simeq K\left(\pi _{n}(X),n\right)}$

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 ${\displaystyle \{X^{n}\}_{n\geq 0}}$ and maps ${\displaystyle f^{n}:X^{n}\to X^{n-1}}$ where

${\displaystyle \pi _{k}\left(X^{n}\right)={\begin{cases}\pi _{k}(X)&k\geq n\\0&{\text{otherwise}}\end{cases}}}$

hence ${\displaystyle X^{0}\simeq X}$. If we take the induced map

${\displaystyle f_{0}^{n+1}:X^{n+1}\to X}$

the homotopy fiber of this map recovers the ${\displaystyle n}$-th postnikov approximation ${\displaystyle X_{n}}$ since the long exact sequence of the fibration

${\displaystyle {\begin{matrix}{\text{Hofiber}}\left(f_{0}^{n+1}\right)&\to &X^{n+1}\\&&\downarrow \\&&X\end{matrix}}}$

we get

${\displaystyle {\begin{matrix}\to &\pi _{k+1}\left({\text{Hofiber}}\left(f_{0}^{n+1}\right)\right)&\to &\pi _{k+1}(X^{n+1})&\to &\pi _{k+1}(X)&\to \\&\pi _{k}\left({\text{Hofiber}}\left(f_{0}^{n+1}\right)\right)&\to &\pi _{k}\left(X^{n+1}\right)&\to &\pi _{k}(X)&\to \\&\pi _{k-1}\left({\text{Hofiber}}\left(f_{0}^{n+1}\right)\right)&\to &\pi _{k-1}\left(X^{n+1}\right)&\to &\pi _{k-1}(X)&\to \end{matrix}}}$

which gives isomorphisms

${\displaystyle \pi _{k-1}\left({\text{Hofiber}}\left(f_{0}^{n+1}\right)\right)\cong \pi _{k}(X)}$

for ${\displaystyle k\leq n}$.

See also[]

• Homotopy cofiber
• Quasi-fibration
• Adams resolution

References[]

• Hatcher, Allen (2002), Algebraic Topology, Cambridge: Cambridge University Press, ISBN 0-521-79540-0.