Homotopy fiber

 ( Algebraic Topology ) 

In mathematics, especially homotopy theory, the homotopy fiber (sometimes called the mapping fiber)Joseph J. Rotman, An Introduction to Algebraic Topology (1988) Springer-Verlag (See Chapter 11 for construction.) is part of a construction that associates a fibration to an arbitrary continuous function of topological spaces $f:A\; \backslash to\; B$. 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

$\backslash cdots\; \backslash to\; \backslash pi\_\{n+1\}(B)\; \backslash to\; \backslash pi\_n(\backslash text\{Hofiber\}(f))\; \backslash to\; \backslash pi\_n(A)\; \backslash to\; \backslash pi\_n(B)\; \backslash to\; \backslash cdots$

Moreover, the homotopy fiber can be found in other contexts, such as homological algebra, where the distinguished triangle

$C(f)\_\backslash bullet-1\; \backslash to\; A\_\backslash bullet\; \backslash to\; B\_\backslash bullet\; \backslash xrightarrow\{+1\}$

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 $f:A\; \backslash to\; B$. 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 $(a,\backslash gamma)$ where $a\; \backslash in\; A$ and $\backslash gamma:I\; \backslash to\; B$ (for $I\; =\; 0,1$) a path such that $\backslash gamma(0)\; =\; f(a)$. We give $E\_f$ a topology by giving it the subspace topology as a subset of $A\backslash times\; B^I$ (where $B^I$ is the space of paths in $B$ which as a function space has the compact-open topology). Then the map $E\_f\; \backslash to\; B$ given by $(a,\backslash gamma)\; \backslash mapsto\; \backslash gamma(1)$ is a fibration. Furthermore, $E\_f$ is homotopy equivalent to $A$ as follows: Embed $A$ as a subspace of $E\_f$ by $a\; \backslash mapsto\; \backslash gamma\_a$ where $\backslash gamma\_a$ 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

which can be defined as the set of all $(a,\backslash gamma)$ with $a\; \backslash in\; A$ and $\backslash gamma:I\; \backslash to\; B$ a path such that $\backslash gamma(0)\; =\; f(a)$ and $\backslash gamma(1)\; =\; *$ for some fixed basepoint $*\; \backslash in\; B$. 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 colimit ^{pg 21} of the diagram

this is because computing the homotopy limit amounts to finding the pullback of the diagram

where the vertical map is the source and target map of a path $\backslash gamma:\; I\; \backslash to\; B$, so

$\backslash gamma\; \backslash mapsto\; (\backslash gamma(0),\; \backslash gamma(1))$

This means the homotopy limit is in the collection of maps

$\backslash left\backslash \{(a,\; \backslash gamma)\; \backslash in\; A\; \backslash times\; B^I\; :\; f(a)\; =\; \backslash gamma(0)\; \backslash text\{\; and\; \}\; \backslash gamma(1)\; =\; *\backslash right\backslash \}$

which is exactly the homotopy fiber as defined above.

If $x\_0$ and $x\_1$ can be connected by a path $\backslash delta$ in $B$, then the diagrams

and

are homotopy equivalent to the diagram

and thus the homotopy fibers of $x\_0$ and $x\_1$ are isomorphic in $\backslash text\{hoTop\}$. 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 $A\; \backslash to\; E\_f$ given above will be a map of fibrations over $B$. This will induce a morphism of their long exact sequences of , from which (by applying the Five Lemma, as is done in the Puppe sequence) one can see that the map 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, A Concise Course in Algebraic Topology, (1999) Chicago Lectures in Mathematics (See chapters 6,7.)

---

Examples

---

Loop space Given a topological space $X$ and the inclusion of a point

$\backslash iota:\; \backslash \{x\_0\backslash \}\; \backslash hookrightarrow\; X$

the homotopy fiber of this map is then

$\backslash left\backslash \{(x\_0,\; \backslash gamma)\; \backslash in\; \backslash \{x\_0\backslash \}\; \backslash times\; X^I\; :\; x\_0\; =\; \backslash gamma(0)\; \backslash text\{\; and\; \}\; \backslash gamma(1)\; =\; x\_0\backslash right\backslash \}$

which is the loop space $\backslash Omega\; X$.

See also: Path space fibration.

---

From a covering space Given a universal covering

$\backslash pi:\backslash tilde\{X\}\; \backslash to\; X$

the homotopy fiber $\backslash text\{Hofiber\}(\backslash pi)$ has the property

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 $\backslash left\backslash \{X\_n\backslash right\backslash \}\_\{n\; \backslash geq\; 0\}$ and maps $f\_n:\; X\_n\; \backslash to\; X\_\{n-1\}$ where

$\backslash pi\_k\backslash left(X\_n\backslash right)\; =\; \backslash begin\{cases\}$

 \pi_k(X) & k \leq n \\
        0 & \text{ otherwise }
     

\end{cases}

and

$X\; \backslash simeq\; \backslash underset\{\backslash leftarrow\}\{\backslash text\{lim\}\}\backslash left(X\_k\backslash right)$

Now, these maps $f\_n$ can be iteratively constructed using homotopy fibers. This is because we can take a map

$X\_\{n-1\}\; \backslash to\; K\backslash left(\backslash pi\_n(X),\; n\; -\; 1\backslash right)$

representing a cohomology class in

$H^\{n-1\}\backslash left(X\_\{n-1\},\; \backslash pi\_n(X)\backslash right)$

and construct the homotopy fiber

$\backslash underset\{\backslash leftarrow\}\{\backslash text\{holim\}\}\backslash left(\backslash begin\{matrix\}$

                          && * \\
                          && \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 $f\_n:\; X\_n\; \backslash to\; X\_\{n-1\}$ is

$\backslash text\{Hofiber\}\backslash left(f\_n\backslash right)\; \backslash simeq\; K\backslash left(\backslash pi\_n(X),\; n\backslash right)$

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 $\backslash \{X^n\backslash \}\_\{n\; \backslash geq\; 0\}$ and maps $f^n:\; X^n\; \backslash to\; X^\{n-1\}$ where

hence $X^0\; \backslash simeq\; X$. If we take the induced map

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

the homotopy fiber of this map recovers the $n$-th postnikov approximation $X\_n$ since the long exact sequence of the fibration

$\backslash begin\{matrix\}$

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

\end{matrix}

we get

$\backslash begin\{matrix\}$

 \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

$\backslash pi\_\{k-1\}\backslash left(\backslash text\{Hofiber\}\backslash left(f^\{n+1\}\_0\backslash right)\backslash right)\; \backslash cong\; \backslash pi\_k(X)$

for $k\; \backslash leq\; n$.

---

See also

• Homotopy cofiber
• Quasi-fibration
• Adams resolution

• .

Post Comment