Homotopy fiber

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.

If ${\displaystyle x_{0}}$ and ${\displaystyle x_{1}}$ can be connected by a path ${\displaystyle \delta }$ in ${\displaystyle B}$, then the diagrams

${\displaystyle {\begin{matrix}&&x_{0}\\&&\downarrow \\A&\xrightarrow {f} &B\end{matrix}}}$

and

${\displaystyle {\begin{matrix}&&x_{1}\\&&\downarrow \\A&\xrightarrow {f} &B\end{matrix}}}$

are homotopy equivalent to the diagram

${\displaystyle {\begin{matrix}&&[0,1]\\&&\downarrow {\delta }\\A&\xrightarrow {f} &B\end{matrix}}}$

and thus the homotopy fibers of ${\displaystyle x_{0}}$ and ${\displaystyle x_{1}}$ are isomorphic in ${\displaystyle {\text{hoTop}}}$. Therefore we often speak about the homotopy fiber of a map without specifying a base point.

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.

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

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 ${\displaystyle \Omega X}$.

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.

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.

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 ${\displaystyle k\leq n}$.