# 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 $f:A\to B$. 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
   $\cdots \to \pi _{n+1}(B)\to \pi _{n}({\text{Hofiber}}(f))\to \pi _{n}(A)\to \pi _{n}(B)\to \cdots$
Moreover, the homotopy fiber can be found in other contexts, such as homological algebra, where the distinguished triangle
   $C(f)_{\bullet }[-1]\to A_{\bullet }\to B_{\bullet }\xrightarrow {[+1]}$
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 $f:A\to B$. If we replace $f$ 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 $E_{f}$ to be the set of pairs $(a,\gamma )$ where $a\in A$ and $\gamma :I\to B$ (for $I=[0,1]$) a path such that $\gamma (0)=f(a)$. We give $E_{f}$ a topology by giving it the subspace topology as a subset of $A\times B^{I}$ (where $B^{I}$ is the space of paths in $B$ which as a function space has the compact-open topology). Then the map $E_{f}\to B$ given by $(a,\gamma )\mapsto \gamma (1)$ is a fibration. Furthermore, $E_{f}$ is homotopy equivalent to $A$ as follows: Embed $A$ as a subspace of $E_{f}$ by $a\mapsto \gamma _{a}$ where $\gamma _{a}$ is the constant path at $f(a)$. Then $E_{f}$ 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
   ${\begin{matrix}{\text{Hofiber}}(f)&\to &E_{f}\\&&\downarrow \\&&B\end{matrix}}$
which can be defined as the set of all $(a,\gamma )$ with $a\in A$ and $\gamma :I\to B$ a path such that $\gamma (0)=f(a)$ and $\gamma (1)=*$ for some fixed basepoint $*\in B$. A consequence of this definition is that if two points of $B$ 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$^{pg21}$ of the diagram
   ${\underset {\leftarrow }{\text{holim}}}\left({\begin{matrix}&&*\\&&\downarrow \\A&\xrightarrow {f} &B\end{matrix}}\right)\simeq F_{f}$
this is because computing the homotopy limit amounts to finding the pullback of the diagram
   ${\begin{matrix}&&B^{I}\\&&\downarrow \\A\times *&\xrightarrow {f} &B\times B\end{matrix}}$
where the vertical map is the source and target map of a path $\gamma :I\to B$, so
   $\gamma \mapsto (\gamma (0),\gamma (1))$
This means the homotopy limit is in the collection of maps
   $\left\{(a,\gamma )\in A\times B^{I}:f(a)=\gamma (0){\text{ and }}\gamma (1)=*\right\}$
which is exactly the homotopy fiber as defined above.
If $x_{0}$ and $x_{1}$ can be connected by a path $\delta$ in $B$, then the diagrams
   ${\begin{matrix}&&x_{0}\\&&\downarrow \\A&\xrightarrow {f} &B\end{matrix}}$
and
   ${\begin{matrix}&&x_{1}\\&&\downarrow \\A&\xrightarrow {f} &B\end{matrix}}$
are homotopy equivalent to the diagram
   ${\begin{matrix}&&[0,1]\\&&\downarrow {\delta }\\A&\xrightarrow {f} &B\end{matrix}}$
and thus the homotopy fibers of $x_{0}$ and $x_{1}$ are isomorphic in ${\text{hoTop}}$. 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 $f$ was a fibration with fiber $F$, then the homotopy equivalence $A\to E_{f}$ given above will be a map of fibrations over $B$. 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.

## Examples
### Loop space

Given a topological space $X$ and the inclusion of a point
   $\iota :\{x_{0}\}\hookrightarrow X$
the homotopy fiber of this map is then
   $\left\{(x_{0},\gamma )\in \{x_{0}\}\times X^{I}:x_{0}=\gamma (0){\text{ and }}\gamma (1)=x_{0}\right\}$
which is the loop space $\Omega X$.
See also: Path space fibration.

### From a covering space

Given a universal covering
   $\pi :{\tilde {X}}\to X$
the homotopy fiber ${\text{Hofiber}}(\pi )$ has the property
   $\pi _{k}({\text{Hofiber}}(\pi ))={\begin{cases}\pi _{1}(X)&k<1\\0&k\geq 1\end{cases}}$
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 $X$, we can construct a sequence of spaces $\left\{X_{n}\right\}_{n\geq 0}$ and maps $f_{n}:X_{n}\to X_{n-1}$ where
   $\pi _{k}\left(X_{n}\right)={\begin{cases}\pi _{k}(X)&k\leq n\\0&{\text{ otherwise }}\end{cases}}$
and
   $X\simeq {\underset {\leftarrow }{\text{lim}}}\left(X_{k}\right)$
Now, these maps $f_{n}$ can be iteratively constructed using homotopy fibers. This is because we can take a map
   $X_{n-1}\to K\left(\pi _{n}(X),n-1\right)$
representing a cohomology class in
   $H^{n-1}\left(X_{n-1},\pi _{n}(X)\right)$
and construct the homotopy fiber
   ${\underset {\leftarrow }{\text{holim}}}\left({\begin{matrix}&&*\\&&\downarrow \\X_{n-1}&\xrightarrow {f} &K\left(\pi _{n}(X),n-1\right)\end{matrix}}\right)\simeq X_{n}$
In addition, notice the homotopy fiber of $f_{n}:X_{n}\to X_{n-1}$ is
   ${\text{Hofiber}}\left(f_{n}\right)\simeq K\left(\pi _{n}(X),n\right)$
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 $\{X^{n}\}_{n\geq 0}$ and maps $f^{n}:X^{n}\to X^{n-1}$ where
   $\pi _{k}\left(X^{n}\right)={\begin{cases}\pi _{k}(X)&k\geq n\\0&{\text{otherwise}}\end{cases}}$
hence $X^{0}\simeq X$. If we take the induced map
   $f_{0}^{n+1}:X^{n+1}\to X$
the homotopy fiber of this map recovers the $n$-th postnikov approximation $X_{n}$ since the long exact sequence of the fibration
   ${\begin{matrix}{\text{Hofiber}}\left(f_{0}^{n+1}\right)&\to &X^{n+1}\\&&\downarrow \\&&X\end{matrix}}$
we get
   ${\begin{matrix}\to &\pi _{k+1}\left({\text{Hofiber}}\left(f_{0}^{n+1}\right)\right)&\to &\pi _{k+1}(X^{n+1})&\to &\pi _{k+1}(X)&\to \\&\pi _{k}\left({\text{Hofiber}}\left(f_{0}^{n+1}\right)\right)&\to &\pi _{k}\left(X^{n+1}\right)&\to &\pi _{k}(X)&\to \\&\pi _{k-1}\left({\text{Hofiber}}\left(f_{0}^{n+1}\right)\right)&\to &\pi _{k-1}\left(X^{n+1}\right)&\to &\pi _{k-1}(X)&\to \end{matrix}}$
which gives isomorphisms
   $\pi _{k-1}\left({\text{Hofiber}}\left(f_{0}^{n+1}\right)\right)\cong \pi _{k}(X)$
for $k\leq n$.