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 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.

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.

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.

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"$X \simeq \underset{\leftarrow}{\text{lim}}\left(X_k\right)$|undefined"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"$f^{n+1}_0: X^{n+1} \to X$"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 $$k \leq n$$.