3.4.6 Excision
Let $X$ be a topological space which is a union of two open subsets $U,V \subseteq X$. Then the diagram
\[ \xymatrix@R =50pt@C=50pt{ U \cap V \ar [r] \ar [d] & U \ar [d] \\ V \ar [r] & X } \]
is a pushout square in the category of topological spaces. Stated more informally, the topological space $X$ can be obtained by gluing $U$ and $V$ along their common open subset $U \cap V$. This observation has a homotopy-theoretic counterpart:
Theorem 3.4.6.1 (Excision). Let $X$ be a topological space, and let $U,V \subseteq X$ be subsets whose interiors $\mathring {U} \subseteq U$ and $\mathring {V} \subseteq V$ comprise an open covering of $X$. Then the diagram of singular simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Sing}_{\bullet }(U \cap V) \ar [r] \ar [d] & \operatorname{Sing}_{\bullet }(U) \ar [d] \\ \operatorname{Sing}_{\bullet }(V) \ar [r] & \operatorname{Sing}_{\bullet }(X) } \]
is a homotopy pushout square (Definition 3.4.2.1).
Warning 3.4.6.3. In the situation of Theorem 3.4.6.1, it is generally not true that the diagram
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Sing}_{\bullet }(U \cap V) \ar [r] \ar [d] & \operatorname{Sing}_{\bullet }(U) \ar [d] \\ \operatorname{Sing}_{\bullet }(V) \ar [r] & \operatorname{Sing}_{\bullet }(X) } \]
is a pushout square of simplicial sets. Concretely, this is because the image of a continuous function $f: | \Delta ^{n} | \rightarrow X$ need not be contained in either $U$ or $V$.
Our goal in this section is to prove a stronger version Theorem 3.4.6.1, where we allow more general coverings of $X$.
Definition 3.4.6.4. Let $X$ be a topological space and let $\operatorname{\mathcal{U}}$ be a collection of subsets of $X$. We say that a singular $n$-simplex $\sigma : | \Delta ^{n} | \rightarrow X$ is $\operatorname{\mathcal{U}}$-small if its image is contained in $U$, for some $U \in \operatorname{\mathcal{U}}$. We let $\operatorname{Sing}_{n}^{\operatorname{\mathcal{U}}}(X)$ denote the subset of $\operatorname{Sing}_{n}(X)$ consisting of the $\operatorname{\mathcal{U}}$-small simplices. Note that the subsets $\{ \operatorname{Sing}_{n}^{\operatorname{\mathcal{U}}}(X) \} _{n \geq 0}$ are stable under the face and degeneracy operators of the simplicial set $\operatorname{Sing}_{\bullet }(X)$, and therefore determine a simplicial subset which we will denote by $\operatorname{Sing}_{\bullet }^{\operatorname{\mathcal{U}}}(X) \subseteq \operatorname{Sing}_{\bullet }(X)$.
Our main result can now be stated as follows:
Theorem 3.4.6.6. Let $X$ be a topological space and let $\operatorname{\mathcal{U}}$ be a collection of subsets of $X$ satisfying $X = \bigcup _{U \in \operatorname{\mathcal{U}}} \mathring {U}$. Then the inclusion map $\operatorname{Sing}_{\bullet }^{\operatorname{\mathcal{U}}}(X) \hookrightarrow \operatorname{Sing}_{\bullet }(X)$ is a weak homotopy equivalence.
Proof of Theorem 3.4.6.1 from Theorem 3.4.6.6.
Let $X$ be a topological space and let $\operatorname{\mathcal{U}}= \{ U, V \} $ be a pair of subsets of $X$. Then $\operatorname{Sing}_{\bullet }^{\operatorname{\mathcal{U}}}(X)$ can be identified with the pushout
\[ \operatorname{Sing}_{\bullet }(U) \coprod _{ \operatorname{Sing}_{\bullet }( U \cap V) } \operatorname{Sing}_{\bullet }(V), \]
formed in the category of simplicial sets. Theorem 3.4.6.6 then asserts that if $X = \mathring {U} \cup \mathring {V}$, then the inclusion
\[ \operatorname{Sing}_{\bullet }(U) \coprod _{ \operatorname{Sing}_{\bullet }( U \cap V) } \operatorname{Sing}_{\bullet }(V) \hookrightarrow \operatorname{Sing}_{\bullet }(X) \]
is a weak homotopy equivalence. By virtue of Remark 3.4.6.2, this is equivalent to Theorem 3.4.6.1.
$\square$
The proof of Theorem 3.4.6.6 is based on the observation that every singular $n$-simplex $\sigma : | \Delta ^ n | \rightarrow X$ can be “decomposed” into $\operatorname{\mathcal{U}}$-small simplices by repeatedly applying the barycentric subdivision described in Proposition 3.3.2.3. To make this precise, we need the following geometric observation:
Lemma 3.4.6.7. Let $V$ be a normed vector space over the real numbers and let $K \subseteq V$ be the convex hull of a finite collection of points $v_0, v_1, \ldots , v_ n \in V$, given by the image of a continuous function:
\[ f: | \Delta ^ n | \rightarrow V \quad \quad (t_0, t_1, \ldots , t_ n) \mapsto t_0 v_0 + t_1 v_1 + \cdots + t_ n v_ n. \]
Let $\sigma $ be any $m$-simplex of the subdivision $\operatorname{Sd}( \Delta ^{n} )$, let $f_{\sigma }$ denote the composite map
\[ | \Delta ^{m} | \xrightarrow { | \sigma | } | \operatorname{Sd}( \Delta ^{n} ) | \simeq | \Delta ^{n} | \xrightarrow {f} V \]
(where the homeomorphism $| \operatorname{Sd}( \Delta ^{n} ) | \leq | \Delta ^ n |$ is supplied by Proposition 3.3.2.3), and let $K_0 \subseteq K$ be the image of $f_{\sigma }$. Then the diameters of $K_0$ and $K$ satisfy the inequality $\mathrm{diam}( K_0 ) \leq \frac{n}{n+1} \mathrm{diam}(K)$.
Proof.
Let us denote the norm on the vector space $V$ by $| \bullet |_{V}$. Fix points $x,y \in | \Delta ^{m} |$; we wish to show that $| f_{\sigma }(x) - f_{\sigma }(y) |_{V} \leq \frac{n}{n+1} \mathrm{diam}(K)$. Note that, if we regard the point $x$ as fixed, then the function $y \mapsto | f_{\sigma }(x) - f_{\sigma }(y) |_{V}$ is convex, and therefore achieves its supremum at some vertex of $| \Delta ^{m} |$. We may therefore assume without loss of generality that $y$ is a vertex of $| \Delta ^{m} |$. Similarly, we may assume that $x$ is a vertex of $| \Delta ^{m} |$. We may also assume that $x \neq y$ (otherwise there is nothing to prove). Exchanging $x$ and $y$ if necessary, it follows that there exist disjoint nonempty subsets $A,B \subseteq \{ 0, 1, \ldots , n \} $ of cardinality $a = |A|$ and $b = |B|$ satisfying
\[ f_{\sigma }(x) = \sum _{i \in A} \frac{ v_ i}{a} \quad \quad f_{\sigma }(y) = \sum _{i \in A \cup B} \frac{ v_ i}{a+b}. \]
We then compute
\begin{eqnarray*} | f_{\sigma }(x) - f_{\sigma }(y) |_{V} & = & | \sum _{(i,j) \in A \times B} \frac{v_ i - v_ j}{a(a+b)} |_{V} \\ & \leq & \sum _{(i,j) \in A \times B} \frac{ | v_ i - v_ j |_{V} }{ a(a+b) } \\ & \leq & \sum _{(i,j) \in A \times B} \frac{ \mathrm{diam}(K) }{ a(a+b) } \\ & = & \frac{b}{a+b} \mathrm{diam}(K) \\ & \leq & \frac{n}{n+1} \mathrm{diam}(K). \end{eqnarray*}
$\square$
Proof of Theorem 3.4.6.6.
Let $X$ be a topological space and let $\operatorname{\mathcal{U}}$ be a collection of subsets of $X$ satisfying $X = \bigcup _{U \in \operatorname{\mathcal{U}}} \mathring {U}$. For each $k \geq 0$, let $Y(k) \subseteq \operatorname{Sing}_{\bullet }(X)$ denote the semisimplicial subset spanned by those singular $n$-simplices $f: | \Delta ^{n} | \rightarrow X$ having the property that, for every $m$-simplex $\sigma $ of the iterated subdivision $\operatorname{Sd}^{k}( \Delta ^ n )$, the composite map
\[ | \Delta ^{m} | \xrightarrow { | \sigma |} | \operatorname{Sd}^{k}( \Delta ^{n} ) | \simeq | \Delta ^{n} | \xrightarrow {f} X \]
is $\operatorname{\mathcal{U}}$-small; here the identification $| \operatorname{Sd}^{k}( \Delta ^ n ) | \simeq | \Delta ^ n |$ is given by iteratively applying the barycentric subdivision of Proposition 3.3.2.3. By construction, we have inclusions of semisimplicial sets
\[ \operatorname{Sing}_{\bullet }^{\operatorname{\mathcal{U}}}(X) = Y(0) \subseteq Y(1) \subseteq Y(2) \subseteq \cdots \subseteq \operatorname{Sing}_{\bullet }(X). \]
We first claim that $\operatorname{Sing}_{\bullet }(X) = \bigcup _{k \geq 0} Y(k)$. Fix a continuous function $f: | \Delta ^{n} | \rightarrow X$, regarded as an $n$-simplex of $\operatorname{Sing}_{\bullet }(X)$; we wish to show that $f$ belongs to $Y(k)$ for $k \gg 0$. Let us identify the topological $n$-simplex $| \Delta ^{n} |$ with the subset of Euclidean space $V = \operatorname{\mathbf{R}}^{n+1}$ given by the convex hull of the standard basis vectors $\{ v_ i \} _{0 \leq i \leq n}$. Then the collection of inverse images $\{ f^{-1}(U) \} _{U \in \operatorname{\mathcal{U}}}$ can be refined to an open covering of $| \Delta ^{n} |$. It follows that there exists a positive real number $\epsilon $ with the property that, for every point $v \in | \Delta ^{n} |$, the open ball
\[ B_{\epsilon }(v) = \{ w \in | \Delta ^{n} |: | v-w|_{V} < \epsilon \} \]
is contained in $f^{-1}(U)$, for some $U \in \operatorname{\mathcal{U}}$. Choose an integer $k$ satisfying $( \frac{n}{n+1} )^{k} \mathrm{diam}( | \Delta ^{n} | ) < \epsilon $. It then follows from iterated application of Lemma 3.4.6.7 that the composite map
\[ | \operatorname{Sd}^{k}( \Delta ^{n} ) | \simeq | \Delta ^{n} | \xrightarrow {f} X \]
carries each simplex of $\operatorname{Sd}^{k}( \Delta ^ n )$ into a subset $U \subseteq X$ belonging to $\operatorname{\mathcal{U}}$, so that $f$ belongs to the semisimplicial subset $Y(k) \subseteq \operatorname{Sing}_{\bullet }(X)$.
Note that the inclusion $\iota : \operatorname{Sing}_{\bullet }^{\operatorname{\mathcal{U}}}(X) \hookrightarrow \operatorname{Sing}_{\bullet }(X)$ is a weak homotopy equivalence of simplicial sets if and only if it is a weak homotopy equivalence when regarded as a morphism of semisimplicial sets (Corollary 3.4.5.5). It follows from the preceding argument that, as a morphism of semisimplicial sets, $\iota $ can be realized as a filtered colimit of the inclusion maps $\iota (k): \operatorname{Sing}_{\bullet }^{\operatorname{\mathcal{U}}}(X) = Y(0) \hookrightarrow Y(k)$. Since the collection of weak homotopy equivalences is closed under filtered colimits (Remark 3.4.5.2), it will suffice to show that each $\iota (k)$ is a weak homotopy equivalence. Proceeding by induction on $k$, we are reduced to showing that each of the inclusion maps $Y(k) \hookrightarrow Y(k+1)$ is a weak homotopy equivalence. Note that the semisimplicial isomorphism $\varphi : \operatorname{Sing}_{\bullet }(X) \simeq \operatorname{Ex}( \operatorname{Sing}_{\bullet }(X) )$ of Example 3.3.2.9 restricts to a map $\varphi ^{\operatorname{\mathcal{U}}}: \operatorname{Sing}_{\bullet }^{\operatorname{\mathcal{U}}}(X) \rightarrow \operatorname{Ex}( \operatorname{Sing}_{\bullet }^{\operatorname{\mathcal{U}}}(X) )$ (which is generally not an isomorphism). Unwinding the definitions, we see that the inclusion $Y(k) \hookrightarrow Y(k+1)$ can be identified with the map $\operatorname{Ex}^{k}( \varphi ^{\operatorname{\mathcal{U}}} ): \operatorname{Ex}^{k}( \operatorname{Sing}_{\bullet }^{\operatorname{\mathcal{U}}}(X) ) \rightarrow \operatorname{Ex}^{k+1}( \operatorname{Sing}_{\bullet }^{\operatorname{\mathcal{U}}}(X) )$ (see Variant 3.3.2.10). By virtue of Corollary 3.4.5.8, it will suffice to show that $\varphi ^{\operatorname{\mathcal{U}}}$ is a weak homotopy equivalence.
Fix an integer $n \geq 0$ as above, let $\operatorname{Chain}[n]$ denote the collection of all nonempty subsets of $[n] = \{ 0 < 1 < \cdots < n \} $. Let $\sigma $ be an $n$-simplex of the simplicial set $\Delta ^{1} \times \operatorname{Sing}_{\bullet }^{\operatorname{\mathcal{U}}}(X)$, which we identify with a pair $(\epsilon , f)$ where $\epsilon : [n] \rightarrow [1]$ is a nondecreasing function and $f: | \Delta ^{n} | \rightarrow X$ is a continuous map of topological spaces. Define a map of sets $g_{\epsilon }: \operatorname{Chain}[n] \rightarrow | \Delta ^{n} |$ by the formula
\[ g_{\epsilon }(S) = \begin{cases} \frac{ \sum _{i \in S} v_{i} }{|S|} & \text{ if } \epsilon |_{S} = 0 \\ v_{ \mathrm{Max}(S)} & \text{ otherwise. } \end{cases} \]
Then $g_{\epsilon }$ extends to a continuous map
\[ \overline{g}_{\epsilon }: | \operatorname{N}_{\bullet }( \operatorname{Chain}[n] ) | \rightarrow | \Delta ^ n | \]
which is affine when restricted to each simplex of $| \operatorname{N}_{\bullet }( \operatorname{Chain}[n] ) | \simeq | \operatorname{Sd}(\Delta ^ n) |$. The composite map
\[ | \operatorname{Sd}( \Delta ^ n) | \xrightarrow { \overline{g}_{\epsilon } } | \Delta ^ n | \xrightarrow {f} X \]
can be identified with an $n$-simplex of $\operatorname{Ex}( \operatorname{Sing}^{\operatorname{\mathcal{U}}}_{\bullet }(X) )$, which we will denote by $h( \sigma )$. It is not difficult to see that the construction $\sigma \mapsto h(\sigma )$ is compatible with face operators, and therefore determines a morphism of semisimplicial sets $h: \Delta ^{1} \times \operatorname{Sing}_{\bullet }^{\operatorname{\mathcal{U}}}(X) \rightarrow \operatorname{Ex}( \operatorname{Sing}_{\bullet }^{\operatorname{\mathcal{U}}}(X) )$. By construction, this morphism fits into a commutative diagram of semisimplicial sets
\[ \xymatrix@C =50pt@R=50pt{ \Delta ^{1} \times \operatorname{Sing}_{\bullet }^{\operatorname{\mathcal{U}}}(X) \ar [dr]^{h} & \{ 0\} \times \operatorname{Sing}_{\bullet }^{\operatorname{\mathcal{U}}}(X) \ar [d]^{ \varphi ^{\operatorname{\mathcal{U}}} } \ar [l]_-{i_0} \\ \{ 1\} \times \operatorname{Sing}_{\bullet }^{\operatorname{\mathcal{U}}}(X) \ar [u]^{i_1} \ar [r]^-{ \rho } & \operatorname{Ex}( \operatorname{Sing}_{\bullet }^{\operatorname{\mathcal{U}}}(X) ), } \]
where $i_0$ and $i_1$ are the inclusion maps and $\rho = \rho _{ \operatorname{Sing}_{\bullet }^{\operatorname{\mathcal{U}}}(X) }$ is the comparison map of Construction 3.3.4.3. Note that the morphisms $i_0$, $i_1$, and $\rho $ are weak homotopy equivalences of simplicial sets (Theorem 3.3.5.1), and therefore also weak homotopy equivalences of semisimplicial sets (Corollary 3.4.5.5). Invoking the two-out-of-three property (Remark 3.4.5.3), we conclude that $h$ and $\varphi ^{\operatorname{\mathcal{U}}}$ are also weak homotopy equivalences of semisimplicial sets.
$\square$