homotopy fiber

In mathematics, especially homotopy theory, the homotopy fiber (sometimes called the mapping fiber)Joseph J. Rotman, An Introduction to Algebraic Topology (1988) Springer-Verlag {{isbn|0-387-96678-1}} (See Chapter 11 for construction.) 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{{Cite web|last=Dugger|first=Daniel|date=|title=A Primer on Homotopy Colimits|url=https://pages.uoregon.edu/ddugger/hocolim.pdf|url-status=live|archive-url=https://web.archive.org/web/20201203225718/https://pages.uoregon.edu/ddugger/hocolim.pdf|archive-date=3 Dec 2020|access-date=|website=}}^{pg 21} 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 {{math|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.J.P. May, [http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf A Concise Course in Algebraic Topology], (1999) Chicago Lectures in Mathematics {{isbn|0-226-51183-9}} (See chapters 6,7.)

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

= 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}

$\text{Hofiber}\left(f_n\right) \simeq K\left(\pi_n(X), n\right)$

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

$f^{n+1}_0: X^{n+1} \to X$

the homotopy fiber of this map recovers the

n

-th postnikov approximation

X_n

since the long exact sequence of the fibration

$\begin{matrix}
\text{Hofiber}\left(f^{n+1}_0\right) & \to & X^{n+1} \\
&& \downarrow \\
&& X
\end{matrix}$

we get

$\begin{matrix}
\to & \pi_{k+1}\left(\text{Hofiber}\left(f^{n+1}_0\right)\right) & \to & \pi_{k+1}(X^{n+1}) & \to & \pi_{k+1}(X) & \to \\
& \pi_{k}\left(\text{Hofiber}\left(f^{n+1}_0\right)\right) & \to & \pi_{k}\left(X^{n+1}\right) & \to & \pi_{k}(X) & \to \\
& \pi_{k-1}\left(\text{Hofiber}\left(f^{n+1}_0\right)\right) & \to & \pi_{k-1}\left(X^{n+1}\right) & \to & \pi_{k-1}(X) & \to
\end{matrix}$

which gives isomorphisms

$\pi_{k-1}\left(\text{Hofiber}\left(f^{n+1}_0\right)\right) \cong \pi_k(X)$

for

k \leq n

References

{{citation| last=Hatcher |first= Allen |title=Algebraic Topology |url=http://pi.math.cornell.edu/~hatcher/AT/ATpage.html |year= 2002 |publisher=Cambridge University Press |place=Cambridge |isbn=0-521-79540-0}}.

Category:Algebraic topology

Category:Homotopy theory