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3.5.6 Serre Fibrations

We now study the counterpart of Definition 3.1.1.1 in the setting of topological spaces.

Definition 3.5.6.1. Let $q: X \rightarrow S$ be a continuous function between topological spaces. We say that $q$ is a Serre fibration if, for every integer $n \geq 0$, every lifting problem

\[ \xymatrix@R =50pt@C=50pt{ \{ 0\} \times | \Delta ^ n | \ar [r] \ar [d] & X \ar [d]^{q} \\ \empty [0,1] \times | \Delta ^ n | \ar [r] \ar@ {-->}[ur] & S } \]

admits a solution.

Example 3.5.6.2. For every topological space $X$, the projection map $X \rightarrow \{ \ast \} $ is a Serre fibration.

Remark 3.5.6.3. Suppose we are given a pullback diagram

\[ \xymatrix@R =50pt@C=50pt{ X' \ar [r] \ar [d]^{q'} & X \ar [d]^{q} \\ S' \ar [r] & S } \]

in the category of topological spaces. If $q$ is a Serre fibration, then $q'$ is also a Serre fibration.

Remark 3.5.6.4. Let $f: X \rightarrow Y$ and $g: Y \rightarrow Z$ be Serre fibrations. Then the composition $(g \circ f): X \rightarrow Z$ is a Serre fibration.

Proposition 3.5.6.5. Let $q: X \rightarrow S$ be a continuous function between topological spaces. Then $q$ is a Serre fibration if and only if the induced map of singular simplicial sets $\operatorname{Sing}_{\bullet }(q): \operatorname{Sing}_{\bullet }(X) \rightarrow \operatorname{Sing}_{\bullet }(S)$ is a Kan fibration.

Remark 3.5.6.6. In the special case where $S$ is a point, Proposition 3.5.6.5 reduces to the assertion that for every topological space $X$, the singular simplicial set $\operatorname{Sing}_{\bullet }(X)$ is a Kan complex, which was established earlier as Proposition 1.1.9.8.

Proof of Proposition 3.5.6.5. Assume first that the map $\operatorname{Sing}_{\bullet }(q): \operatorname{Sing}_{\bullet }(X) \rightarrow \operatorname{Sing}_{\bullet }(S)$ is a Kan fibration of simplicial sets. It follows that, for each $n \geq 0$, $\operatorname{Sing}_{\bullet }(q)$ has the right lifting property with respect to the inclusion map $\{ 0\} \times \Delta ^ n \hookrightarrow \Delta ^1 \times \Delta ^ n$ (which is anodyne, by virtue of Proposition 3.1.2.8). It follows that the continuous function $q$ has the right lifting property with respect to the induced map of geometric realizations $| \{ 0\} \times \Delta ^ n | \hookrightarrow | \Delta ^1 \times \Delta ^ n |$, which can be identified with the inclusion $\{ 0\} \times | \Delta ^ n | \hookrightarrow [0,1] \times | \Delta ^ n |$ (see Corollary 3.5.2.2). Allowing $n$ to vary, we deduce that $q$ is a Serre fibration.

We now prove the converse. Suppose that $q$ is a Serre fibration; we wish to show that the induced map of simplicial sets $\operatorname{Sing}_{\bullet }(q): \operatorname{Sing}_{\bullet }(X) \rightarrow \operatorname{Sing}_{\bullet }(S)$ has the right lifting property with respect to the horn inclusion $\Lambda ^{n}_{i} \hookrightarrow \Delta ^ n$ for every pair of integers $0 \leq i \leq n$ with $n > 0$. Equivalently, we wish to show that $q$ has the right lifting property with respect to the inclusion of geometric realizations $\iota : | \Lambda ^{n}_{i} | \hookrightarrow | \Delta ^ n |$. We proceed by refining the proof of Proposition 1.1.9.8. Define a continuous function $c: | \Delta ^{n} | \rightarrow [0,1]$ by the formula $c( t_0, t_1, \cdots , t_ n) = \min \{ t_0, \ldots , t_{i-1}, t_{i+1}, \ldots , t_ n \} $. Let $h: [0,1] \times | \Delta ^ n | \rightarrow | \Delta ^ n |$ be the continuous function given by the formula

\[ h( s, (t_0, \cdots , t_ n) ) = (t_0 - \lambda , \cdots , t_{i-1} - \lambda , t_{i} + n \lambda , t_{i+1} - \lambda , \cdots , t_ n - \lambda ) \]

\[ \lambda = \max \{ 0, c(t_0, \cdots , t_ n) - s \} \]

By construction, the composition

\[ | \Delta ^{n} | \xrightarrow { (c,\operatorname{id})} [0,1] \times | \Delta ^ n | \xrightarrow {h} | \Delta ^ n | \]

is the identity map. Moreover, the function $(c,\operatorname{id})$ carries the horn $| \Lambda ^{n}_{i} | \subset | \Delta ^ n |$ to the closed subset $\{ 0\} \times | \Delta ^ n | \subseteq | \Delta ^{n} |$, and the function $h$ carries $\{ 0\} \times | \Delta ^ n |$ to the horn $| \Lambda ^{n}_{i} | \subset | \Delta ^ n |$. It follows that $h$ and $(c,\operatorname{id})$ exhibit $\iota $ as a retract of the inclusion map $\iota ': \{ 0\} \times | \Delta ^ n | \hookrightarrow [0,1] \times | \Delta ^ n |$ in the category of topological spaces. Consequently, to show that $q$ has the right lifting property with respect to $\iota $, it will suffice to show that it has the right lifting property with respect to $\iota '$ (Proposition 1.4.4.9), which follows immediately from our assumption that $q$ is a Serre fibration. $\square$

Exercise 3.5.6.7. Show that, for every pair of integers $0 \leq i \leq n$ with $n > 0$, there exists a homeomorphism of topological spaces

\[ h: [0,1] \times | \Delta ^{n-1} | \simeq | \Delta ^{n} | \]

which restricts to a homeomorphism of $\{ 0\} \times | \Delta ^{n-1} |$ with the horn $| \Lambda ^{n}_{i} | \subset | \Delta ^ n |$. Use this homeomorphism to give a more direct proof of Proposition 3.5.6.5.

Corollary 3.5.6.8 (The Homotopy Extension Lifting Property). Let $q: X \rightarrow S$ be a continuous function between topological spaces. The following conditions are equivalent:

$(1)$

The morphism $q$ is a Serre fibration.

$(2)$

For every simplicial set $B$, every lifting problem

\[ \xymatrix@R =50pt@C=50pt{ [0,1] \times |B| \ar [r] \ar@ {^{(}->}[d] & X \ar [d]^{q} \\ \{ 0\} \times |B| \ar [r] \ar@ {-->}[ur] & S } \]

admits a solution.

$(3)$

For every monomorphism of simplicial sets $A \hookrightarrow B$, every lifting problem

3.68
\begin{equation} \begin{gathered}\label{equation:Serre-fibration-HELP} \xymatrix@R =50pt@C=50pt{ ( [0,1] \times |A| ) \coprod _{ (\{ 0\} \times |A|) } ( \{ 0\} \times |B|) \ar [r] \ar@ {^{(}->}[d] & X \ar [d]^{q} \\ \empty [0,1] \times | B | \ar [r] \ar@ {-->}[ur] & S } \end{gathered} \end{equation}

admits a solution.

Proof. The implication $(3) \Rightarrow (2) \Rightarrow (1)$ are immediate from the definition. We will complete the proof by showing that $(1)$ implies $(3)$. Using Corollary 3.5.2.2, we observe that every lifting problem of the form (3.68) can be rewritten as a lifting problem

\[ \xymatrix@R =50pt@C=50pt{ (\Delta ^1 \times _{A} ) \coprod _{ ( \{ 0\} \times A) } ( \{ 0 \} \times B) \ar [r] \ar [d]^{\iota } & \operatorname{Sing}_{\bullet }(X) \ar [d]^{ \operatorname{Sing}_{\bullet }(q) } \\ \Delta ^1 \times B \ar [r] \ar@ {-->}[ur] & \operatorname{Sing}_{\bullet }(S) } \]

in the category of simplicial sets. If $q$ is Serre fibration, then $\operatorname{Sing}_{\bullet }(q)$ is a Kan fibration (Proposition 3.5.6.5), so the existence of the desired lifting follows from the observation that $\iota $ is an anodyne morphism (Proposition 3.1.2.8). $\square$

Remark 3.5.6.9. A continuous function $q: X \rightarrow S$ is a Hurewicz fibration if, for every topological space $Y$, every lifting problem

\[ \xymatrix@R =50pt@C=50pt{ \{ 0\} \times Y \ar [r] \ar [d] & X \ar [d]^{q} \\ \empty [0,1] \times Y \ar [r] \ar@ {-->}[ur] & S } \]

admits a solution. Equivalently, $q$ is a Hurewicz fibration if the evaluation map

\[ \operatorname{Hom}_{\operatorname{Top}}( [0,1], X) \rightarrow \operatorname{Hom}_{\operatorname{Top}}( \{ 0\} , X) \times _{ \operatorname{Hom}_{\operatorname{Top}}( \{ 0\} , S) } \operatorname{Hom}_{\operatorname{Top}}( [0,1], S) \]

admits a continuous section, where we endow $\operatorname{Hom}_{\operatorname{Top}}( [0,1], X)$ and $\operatorname{Hom}_{\operatorname{Top}}( [0,1], S)$ with their compact-open topologies. Every Hurewicz fibration is a Serre fibration. However, the converse is false.

The lifting condition of Definition 3.5.6.1 can be tested locally:

Proposition 3.5.6.10. Let $q: X \rightarrow S$ be a continuous function between topological spaces. Suppose that, for every point $s \in S$, there exists an open subset $U \subseteq S$ containing the point $s$ for which the induced map $q_{U}: U \times _{S} X \rightarrow U$ is a Serre fibration. Then $q$ is a Serre fibration.

Proof. Let $\operatorname{\mathcal{U}}$ be the collection of all open subsets $U \subseteq S$ for which the map $q_{U}$ is a Serre fibration. Suppose we are given a finite simplicial set $B$ and a simplicial subset $A \subseteq B$. We will say that a lifting problem

\[ \xymatrix@R =50pt@C=50pt{ ( [0,1] \times |A| ) \coprod _{ (\{ 0\} \times |A|) } ( \{ 0\} \times |B|) \ar [r] \ar@ {^{(}->}[d]^{\iota } & X \ar [d]^{q} \\ \empty [0,1] \times | B | \ar [r]^-{h} \ar@ {-->}[ur] & S } \]

is $\operatorname{\mathcal{U}}$-small if, for every element $s \in [0,1]$ and every simplex $\sigma : \Delta ^{k} \rightarrow B$, the image of the composite map

\[ \{ s\} \times | \Delta ^{k} | \xrightarrow {\sigma } [0,1] \times |B| \xrightarrow {h} S \]

is contained in some open set belonging to the cover $\operatorname{\mathcal{U}}$. We first claim that every $\operatorname{\mathcal{U}}$-small lifting problem admits a solution. Proceeding by induction on the number of simplices of $B$ which do not belong to $A$, we can reduce to the case where $B$ is a standard simplex and $A$ is its boundary. In this case, it follows from our $\operatorname{\mathcal{U}}$-smallness assumption and the compactness of the product $[0,1] \times | B |$ that there exists some integer $m \gg 0$ with the property that, for each $1 \leq k \leq m$, the composite map

\[ [ \frac{ k-1}{m}, \frac{k}{m} ] \times | B | \hookrightarrow [0,1] \times | B | \xrightarrow {h} S \]

has image contained in some open set $U_{k} \in \operatorname{\mathcal{U}}$. Writing $\iota $ as a composition of inclusion maps

\[ ( [0,1] \times |A| ) \coprod _{ ([0, \frac{k-1}{m}] \times |A|) } ( [0, \frac{k-1}{m}] \times |B|) \hookrightarrow ( [0,1] \times |A| ) \coprod _{ ([0, \frac{k}{m}] \times |A|) } ( [0, \frac{k}{m}] \times |B|), \]

we are reduced to solving a finite sequence of lifting problems

\[ \xymatrix@R =50pt@C=50pt{ ( [ \frac{k-1}{m}, \frac{k}{m} ] \times |A| ) \coprod _{ (\{ \frac{k-1}{m} \} \times |A|) } ( \{ \frac{k-1}{m} \} \times |B|) \ar [r] \ar@ {^{(}->}[d] & U_ k \times _{S} X \ar [d]^{q_{U_ k}} \\ \empty [ \frac{k-1}{m}, \frac{k}{m} ] \times | B | \ar [r] \ar@ {-->}[ur] & U_ k, } \]

which is possible by virtue of our assumption that $q_{U_ k}$ is a Serre fibration (Corollary 3.5.6.8).

Fix an integer $n \geq 0$; we wish to show that every lifting problem

3.69
\begin{equation} \begin{gathered}\label{equation:Serre-fibration-local} \xymatrix@R =50pt@C=50pt{ \{ 0\} \times | \Delta ^ n | \ar [r] \ar [d] & X \ar [d]^{q} \\ \empty [0,1] \times | \Delta ^ n | \ar [r]^-{h} \ar@ {-->}[ur] & S } \end{gathered} \end{equation}

admits a solution. Fix an integer $t \geq 0$, and $B = \operatorname{Sd}^{t}( \Delta ^{n} )$ denote the $t$-fold subdivision of $\Delta ^ n$. Then Proposition 3.3.3.6 supplies a homeomorphism $| B | \simeq | \Delta ^ n |$, which we can use to rewrite (3.69) as a lifting problem

3.70
\begin{equation} \begin{gathered}\label{equation:Serre-fibration-local2} \xymatrix@R =50pt@C=50pt{ \{ 0\} \times | B | \ar [r] \ar [d] & X \ar [d]^{q} \\ \empty [0,1] \times | B| \ar [r]^-{h'} \ar@ {-->}[ur] & S } \end{gathered} \end{equation}

It follows from Lemma 3.4.6.7 that the lifting problem (3.70) is $\operatorname{\mathcal{U}}$-small for $t \gg 0$, and therefore admits a solution by the first step of the proof. $\square$

Corollary 3.5.6.11. Let $q: X \rightarrow S$ be a continuous function between topological spaces. Suppose that $q$ is a fiber bundle: that is, for every point $s \in S$, there exists an open set $U \subseteq S$ containing $s$ and a homeomorphism $U \times _{S} X \simeq U \times Y$ for some topological space $Y$ (compatible with the projection to $U$). Then $q$ is a Serre fibration.

Proof. By virtue of Proposition 3.5.6.10, it suffices to check this locally on $S$ and we may therefore assume that there exists a pullback diagram

\[ \xymatrix@R =50pt@C=50pt{ X \ar [d]^{q} \ar [r] & Y \ar [d] \\ S \ar [r] & \{ \ast \} } \]

for some topological space $Y$. Using Remark 3.5.6.3, we are reduced to showing that the projection map $Y \rightarrow \{ \ast \} $ is a Serre fibration, which follows from Example 3.5.6.2. $\square$