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 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^[2]^{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 $F \to F f$ 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 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 = 0 \\ 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^{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

^ Joseph J. Rotman, An Introduction to Algebraic Topology (1988) Springer-Verlag ISBN 0-387-96678-1 (See Chapter 11 for construction.)
^ Dugger, Daniel. "A Primer on Homotopy Colimits". Archived 3 Dec 2020 at the Wayback Machine.
^ J.P. May, A Concise Course in Algebraic Topology, (1999) Chicago Lectures in Mathematics ISBN 0-226-51183-9 (See chapters 6,7.)

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