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3.4.7 The Seifert van-Kampen Theorem

Let $X$ be a topological space containing a pair of subsets $U,V \subseteq X$. If $X$ is covered by the interiors $\mathring {U}$ and $\mathring {V}$, then Theorem 3.4.6.1 guarantees that the diagram of Kan complexes

\[ \xymatrix { \operatorname{Sing}_{\bullet }(U \cap V) \ar [r] \ar [d] & \operatorname{Sing}_{\bullet }(U) \ar [d] \\ \operatorname{Sing}_{\bullet }(V) \ar [r] & \operatorname{Sing}_{\bullet }(X) } \]

is homotopy coCartesian. In this section, we apply this assertion of recover several classical results in algebraic topology.

Theorem 3.4.7.1 (Seifert-van Kampen). Let $X$ be a topological space containing a pair of subsets $U,V \subseteq X$ which satisfy the following conditions:

$(1)$

The topological spaces $U$, $V$, and $U \cap V$ are path connected.

$(2)$

The interiors $\mathring {U} \subseteq U$ and $\mathring {V} \subseteq V$ comprise an open covering of $X$.

Then, for every point $x \in U \cap V$, the diagram

\[ \xymatrix { \pi _{1}( U \cap V, x) \ar [r] \ar [d] & \pi _{1}(U,x) \ar [d] \\ \pi _{1}(V,x) \ar [r] & \pi _{1}(X,x) } \]

is a pushout square in the category of groups.

We will deduce Theorem 3.4.7.1 from the following variant of Brown ([MR0220279]), which does not require any connectivity hypotheses.

Theorem 3.4.7.2 (Seifert-van Kampen, Groupoid Version). 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 fundamental groupoids

\[ \xymatrix { \pi _{\leq 1}(U \cap V) \ar [r] \ar [d] & \pi _{\leq 1}(U) \ar [d] \\ \pi _{\leq 1}(V) \ar [r] & \pi _{\leq 1}(X) } \]

is a pushout square in the (ordinary) category $\operatorname{Cat}$.

Proof. Let $\operatorname{\mathcal{C}}$ be a category; we wish to show that the diagram of sets $\sigma :$

\[ \xymatrix { \operatorname{Hom}_{\operatorname{Cat}}( \pi _{\leq 1}(U \cap V), \operatorname{\mathcal{C}}) & \operatorname{Hom}_{\operatorname{Cat}}( \pi _{\leq 1}(U), \operatorname{\mathcal{C}}) \ar [l] \\ \operatorname{Hom}_{\operatorname{Cat}}( \pi _{\leq 1}(V), \operatorname{\mathcal{C}}) \ar [u] & \operatorname{Hom}_{\operatorname{Cat}}( \pi _{\leq 1}(X), \operatorname{\mathcal{C}}) \ar [u] \ar [l] } \]

is a pullback square. Replacing $\operatorname{\mathcal{C}}$ by its core $\operatorname{\mathcal{C}}^{\simeq }$ (Construction 1.2.4.4), we may assume without loss of generality that $\operatorname{\mathcal{C}}$ is a groupoid. Let $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ denote the nerve of $\operatorname{\mathcal{C}}$, so that we can identify $\sigma $ with the diagram

\[ \xymatrix { \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{Sing}_{\bullet }(U \cap V), \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) ) & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{Sing}_{\bullet }(U), \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) ) \ar [l] \\ \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{Sing}_{\bullet }(V), \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) ) \ar [u] & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{Sing}_{\bullet }(X), \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) ). \ar [u] \ar [l] } \]

Let $K$ denote the pushout $\operatorname{Sing}_{\bullet }(U) \coprod _{ \operatorname{Sing}_{\bullet }(U \cap V) } \operatorname{Sing}_{\bullet }(V)$, which we regard as a simplicial subset of $\operatorname{Sing}_{\bullet }(X)$. Unwinding the definitions, we must show that every morphism of simplicial sets $f: K \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ extends uniquely to a map $\overline{f}: \operatorname{Sing}_{\bullet }(X) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$. Note that the inclusion $K \hookrightarrow \operatorname{Sing}_{\bullet }(X)$ is a weak homotopy equivalence (Theorem 3.4.6.1) and therefore anodyne (Corollary 3.3.7.5), so the existence of $\overline{f}$ follows from the observation that $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is a Kan complex (Proposition 1.2.4.2). To prove uniqueness, suppose that we are given a pair of maps $\overline{f}, \overline{f}': \operatorname{Sing}_{\bullet }(X) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ satisfying $\overline{f}|_{K} = f = \overline{f}'|_{K}$. It follows that there exists a homotopy $h: \Delta ^1 \times \operatorname{Sing}_{\bullet }(X) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ which is constant when restricted to $\Delta ^1 \times K$. Note that $\overline{f}$ and $\overline{f}'$ can be identified with functors $F,F': \pi _{\leq 1}(X) \rightarrow \operatorname{\mathcal{C}}$, and $h$ with a natural transformation of functors $H: F \rightarrow F'$. Since every vertex of $\operatorname{Sing}_{\bullet }(X)$ is contained in $K$, this natural transformation carries each point $x \in X$ to the identity morphism $\operatorname{id}_{ \overline{f}(x) }: F(x) \rightarrow F(x) = F'(x)$. It follows that the functors $F$ and $F'$ are identical, so that the morphisms $\overline{f}$ and $\overline{f}'$ are the same. $\square$

Proof of Theorem 3.4.7.1. For every group $G$, let us write $BG$ for the groupoid having a single object with automorphism group $G$ (Example 1.2.4.3). Fix a point $x \in U \cap V$. To show that the diagram

\[ \xymatrix { \pi _{1}( U \cap V, x) \ar [r] \ar [d] & \pi _{1}(U,x) \ar [d] \\ \pi _{1}(V,x) \ar [r] & \pi _{1}(X,x) } \]

is a pushout square in the category of groups, it will suffice to show that the diagram $\sigma _0$:

\[ \xymatrix { B \pi _{1}( U \cap V, x) \ar [r] \ar [d] & B \pi _{1}(U,x) \ar [d] \\ B\pi _{1}(V,x) \ar [r] & B \pi _{1}(X,x) } \]

is a pushout square in the (ordinary) category $\operatorname{Cat}$.

For each point $y \in X$, choose a continuous path $p_{y}: [0,1] \rightarrow X$ satisfying $p(0) = x$ and $p(1) = y$. By virtue of our assumption that $U$, $V$, and $U \cap V$ are path connected, we can arrange that these paths satisfy the following requirements:

  • If $y = x$, then $p_ y: [0,1] \rightarrow X$ is the constant map taking the value $x$.

  • If $y$ is contained in the intersection $U \cap V$, then the path $p_{y}$ factors through $U \cap V$.

  • If $y$ is contained in $U$, then the path $p_{y}$ factors through $U$.

  • If $y$ is contained in $V$, then the path $p_{y}$ factors through $V$.

Note that, for $W \in \{ X, U, V, U \cap V \} $, we can identify $B \pi _{1}(W,x)$ with the full subcategory of $\pi _{\leq 1}(W)$ spanned by the point $x$. Let $r_{W}: \pi _{\leq 1}(W) \rightarrow B \pi _{1}(W,x)$ be the functor which carries each point of $W$ to the point $x$, and each morphism $\alpha \in \operatorname{Hom}_{\pi _{\leq 1}(W)}( y, z )$ to the composition $[p_{z}]^{-1} \circ \alpha \circ [p_{y}]$ (where $[p_ y]$ and $[p_ z]$ denote the homotopy classes of the paths $p_ y$ and $p_ z$, regarded as morphisms in the fundamental groupoid $\pi _{\leq 1}(W)$). The functors $r_{W}$ restrict to the identity on $B \pi _{1}(W,x)$ and are compatible as $W$ varies, and therefore exhibit $\sigma _0$ as a retract of the diagram $\sigma :$

\[ \xymatrix { \pi _{\leq 1}(U \cap V) \ar [r] \ar [d] & \pi _{\leq 1}(U) \ar [d] \\ \pi _{\leq 1}(V) \ar [r] & \pi _{\leq 1}(X) } \]

in the category $\operatorname{Fun}( [1] \times [1], \operatorname{Cat})$. Since $\sigma $ is a pushout square (by virtue of Theorem 3.4.7.2), it follows that $\sigma _0$ is also a pushout square. $\square$

If $X$ is a topological space and $U \subseteq X$ is a subspace (not necessarily open), we will write $\mathrm{H}_{\ast }(X,U; \operatorname{\mathbf{Z}})$ for the relative homology groups of the pair $(X,U)$: that is, the homology groups of the quotient chain complex $\mathrm{C}_{\ast }(X; \operatorname{\mathbf{Z}}) / \mathrm{C}_{\ast }(U;\operatorname{\mathbf{Z}})$ (see Example 2.5.5.3).

Theorem 3.4.7.3 (Excision for Homology). 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 inclusion $U \hookrightarrow X$ induces an isomorphism of relative homology groups

\[ \mathrm{H}_{\ast }( U, U \cap V; \operatorname{\mathbf{Z}}) \rightarrow \mathrm{H}_{\ast }(X, V; \operatorname{\mathbf{Z}}). \]

Proof. Let $K$ denote the pushout $\operatorname{Sing}_{\bullet }(U) \coprod _{ \operatorname{Sing}_{\bullet }(U \cap V) } \operatorname{Sing}_{\bullet }(V)$. We then have a commutative diagram of short exact sequences of chain complexes

\[ \xymatrix { 0 \ar [r] & \mathrm{C}_{\ast }(V; \operatorname{\mathbf{Z}}) \ar@ {=}[d] \ar [r] & \mathrm{C}_{\ast }(K; \operatorname{\mathbf{Z}}) \ar [r] \ar [d]^{\theta '} & \mathrm{C}_{\ast }(U;\operatorname{\mathbf{Z}}) / \mathrm{C}_{\ast }(U \cap V;\operatorname{\mathbf{Z}}) \ar [d]^{\theta } \ar [r] & 0 \\ 0 \ar [r] & \mathrm{C}_{\ast }(V; \operatorname{\mathbf{Z}}) \ar [r] & \mathrm{C}_{\ast }(X; \operatorname{\mathbf{Z}}) \ar [r] & \mathrm{C}_{\ast }(X; \operatorname{\mathbf{Z}}) / \mathrm{C}_{\ast }(V; \operatorname{\mathbf{Z}}) \ar [r] & 0. } \]

Consequently, to show that $\theta $ is a quasi-isomorphism, it will suffice to show that $\theta '$ is a quasi-isomorphism (Remark 2.5.1.7). This is a special case of Proposition 3.1.5.14, since the inclusion $K \hookrightarrow \operatorname{Sing}_{\bullet }(X)$ is a weak homotopy equivalence of simplicial sets (Theorem 3.4.6.1). $\square$

Remark 3.4.7.4 (The Mayer-Vietoris Sequence). Let $X$ be a topological space, let $U,V \subseteq X$ be subsets whose interiors $\mathring {U} \subseteq U$ and $\mathring {V} \subseteq V$ comprise an open covering of $X$, and set $K = \operatorname{Sing}_{\bullet }(U) \coprod _{ \operatorname{Sing}_{\bullet }(U \cap V) } \operatorname{Sing}_{\bullet }(V)$. Then the inclusion $K \hookrightarrow \operatorname{Sing}_{\bullet }(X)$ induces a quasi-isomorphism $\mathrm{C}_{\ast }(K; \operatorname{\mathbf{Z}}) \hookrightarrow \mathrm{C}_{\ast }(X; \operatorname{\mathbf{Z}})$ (by virtue of Theorem 3.4.6.1 and Proposition 3.1.5.14), and we have a short exact sequence of chain complexes

\[ 0 \rightarrow \mathrm{C}_{\ast }(U \cap V; \operatorname{\mathbf{Z}}) \rightarrow \mathrm{C}_{\ast }(U; \operatorname{\mathbf{Z}}) \oplus \mathrm{C}_{\ast }(V; \operatorname{\mathbf{Z}}) \rightarrow \mathrm{C}_{\ast }(K; \operatorname{\mathbf{Z}}) \rightarrow 0. \]

Passing to homology groups (see Construction ), we obtain a long exact sequence of abelian groups

\[ \cdots \rightarrow \mathrm{H}_{\ast +1}(X; \operatorname{\mathbf{Z}}) \xrightarrow {\delta } \mathrm{H}_{\ast }(U \cap V; \operatorname{\mathbf{Z}}) \rightarrow \mathrm{H}_{\ast }(U; \operatorname{\mathbf{Z}}) \oplus \mathrm{H}_{\ast }(V; \operatorname{\mathbf{Z}}) \rightarrow \mathrm{H}_{\ast }(X; \operatorname{\mathbf{Z}}) \rightarrow \cdots \]

which we refer to as the Mayer-Vietoris sequence of the covering $\{ U,V \} $. The existence of this sequence is essentially equivalent to the statement of Theorem 3.4.7.3.