Skip to main content

Math 344-1,2: Kursobjekt

Aaron Greicius

Section 2.15 Seifert-van Kampen theorem

Definition 2.15.1. Pushouts of groups.

A pushout (or fibered coproduct) of a pair of group homomorphisms \(g_1\colon G_0\rightarrow G_1\) and \(g_2\colon G_0\rightarrow G_2\) is a group \(G\) equipped with homomorphisms \(f_1\colon G_1\rightarrow G\) and \(f_2\colon G_2\rightarrow G\) satisfying \(f_1\circ g_1=f_2\circ g_2\) and the following universal mapping property of pushouts:
  • Universal mapping property of pushouts.
    If \(H\) is a group and \(\phi_1\colon G_1\rightarrow H\) and \(\phi_2\colon G_2\rightarrow H\) are homomorphisms satisfying
    \begin{equation*} \phi_1\circ g_1=\phi_2\circ g_2\text{,} \end{equation*}
    then there is a unique homomorphism \(\phi\colon G\rightarrow H\) satisfying
    \begin{equation*} \phi_k=\phi\circ f_k \end{equation*}
    for \(1\leq k\leq 2\text{.}\)
Pushout commutative diagram
As we will see, a pushout \(G\) is unique up to isomorphism. We write \(G_1\sqcup_{G_0} G_2\) to denote a pushout of the maps \(g_1\colon G_0\rightarrow G_1\) and \(g_2\colon G_0\rightarrow G_2\text{.}\)

Proof.

Homework exercise.

Proof.

First we introduce some useful notation. For a path \(f\in P(X)\text{,}\) we write \([f]\) to denote the path homotopy equivalence class of \(f\) in \(X\text{;}\) if \(f\in P(W)\subseteq P(X)\) for \(W\in \{U, V, U\cap V\}\text{,}\) then we write \([f]_W\) to denote the path homotopy equivalance class of \(f\) in \(W\text{.}\) We can use this notation to describe our maps \(i_1,i_2,j_1,j_2\) as follows:
\begin{align*} i_1([f]_{U\cap V}) \amp = [f]_{U} \amp i_2([f]_{U\cap V}) \amp =[f]_{V} \amp (f\in P(U\cap V))\\ j_1([f]_U) \amp = [f] \amp \amp \amp (f\in P(U)) \\ j_2([f]_V)\amp =[f] \amp \amp \amp (f\in P(V)) \text{.} \end{align*}

Uniqueness of \(\phi\).

We can see by the weak Seifert-van Kampen theorem that there is at most one \(\phi\) satisfying the given conditions. Indeed, given any \([f]\in \pi_1(X,x_0)\text{,}\) we have
\begin{equation*} [f]=[g_1]*[g_2]*\cdots *[g_n]\text{,} \end{equation*}
where \([g_k]=j_1([g_k]_U)\) or \([g_k]=j_2([g_k]_V)\) for all \(1\leq k\leq n\text{.}\) But then since \(\phi\) is a group homomorphism, we must have
\begin{align*} \phi([f]) \amp =\phi([g_1])*\phi([g_2])*\cdots *\phi([g_n]) \end{align*}
where
\begin{equation*} \phi([g_k])=\begin{cases} \phi(j_1([g_k]_U))=\phi_1([g_k]_U) \amp \text{if } g_k\in P(U; x_0,x_0) \\ \phi(j_2([g_k]_V))=\phi_2([g_k]_V) \amp \text{if } g_k\in P(V; x_0,x_0). \end{cases} \end{equation*}
Thus the homomorphism \(\phi\) is uniquely determined by the homomorphisms \(\phi_1\colon \pi_1(U,x_0)\rightarrow H\) and \(\phi_2\colon \pi_1(V,x_0)\rightarrow H\text{.}\)

Existence of \(\phi\).

We will define \(\phi\) by constructing a function \(\tau\colon P(X)\rightarrow H\) that satisfies the following conditions:
\begin{align} [f]=[g] \amp \implies \tau([f])=\tau([g])\tag{2.15.1}\\ \tau(f*g)\amp =\tau(f)*\tau(g) \tag{2.15.2}\\ \tau(f)\amp =\phi_1([f]_U) \text{ for all } f\in P(U; x_0,x_0) \tag{2.15.3}\\ \tau(f)\amp =\phi_2([f]_V) \text{ for all } f\in P(V; x_0,x_0) \text{.}\tag{2.15.4} \end{align}
With such a \(\tau\) in hand, we let \(\phi\colon \pi_1(X,x_0)\rightarrow H\) be defined as \(\phi([f])=\tau(f)\text{.}\) It is clear from properties (2.15.1)–(2.15.2) that \(\phi\) is a homomorphism; and properties (2.15.3)–(2.15.4) guarantee that \(\phi\circ j_k=\phi_k\) for \(1\leq k\leq 2\text{.}\)
Our construction of \(\tau\) proceeds by building up the domain of definition of \(\tau\) step by step: from \(D_1=P(U; x_0,x_0)\cup P(V; x_0,x_0)\) to \(D_2=P(U)\cup P(V)\) to \(D=P(X)\text{.}\)

Construction: \(\tau\vert_{D_1}\text{,}\) \(D_1=P(U; x_0,x_0)\cup P(V; x_0,x_0)\).

If \(f\) is a path based on \(x_0\) lying either in \(U\) or \(V\text{,}\) we define
\begin{equation*} \tau\vert_{D_1}([f])=\begin{cases} \phi_1([f]_U)\amp \text{if } f\in P(U;x_0,x_0)\\ \phi_2([f]_V)\amp \text{if } f\in P(V; x_0,x_0). \end{cases} \end{equation*}
Observe that \(\tau\vert_{D_1}\) satisfies the following relative versions of properties (2.15.1)–(2.15.2):
\begin{align} [f]_W=[g]_W \amp \implies \tau\vert_{D_1}(f)=\tau\vert_{D_1}(g) \text{ for all } f,g\in P(W;x_0,x_0)\tag{2.15.5}\\ \tau\vert_{D_1}(f*g) \amp = \tau\vert_{D_1}(f)\tau\vert_{D_1}(g) \text{ for all } f,g\in P(W; x_0,x_0)\text{,}\tag{2.15.6} \end{align}
where \(W=U\) or \(W=V\text{.}\)

Construction: \(\tau\vert_{D_2}\text{,}\) \(D_2=P(U)\cup P(V)\).

First we fix a family of paths \(\{\alpha_x\}_{x\in X}\) satisfying the following conditions:
  1. \(\alpha_x\in P(X; x_0, x)\) (i.e., \(\alpha_x\) is a path from \(x_0\) to \(x\)),
  2. \(\alpha_{x_0}=e_{x_0}\text{,}\)
  3. \(x\in W\implies \alpha_x\in P(W; x_0, x)\) for \(W\in \{U, V, U\cap V\}\text{.}\)
This is where we use the condition that \(U\text{,}\) \(V\text{,}\) and \(U\cap V\) are all path connected (and that \(X=U\cup V\)).
Next, given a path (not necessarily a loop) \(f\in P(W)\text{,}\) where \(W=U\) or \(W=V\text{,}\) we define \(\tau\vert_{D_2}(f)=\tau\vert_{D_1}(\alpha_x*f*\overline{\alpha}_y)\text{.}\) Note that the right-hand side here makes sense since \(\alpha_x*f*\overline{\alpha}_y\in P(W; x_0,x_0\text{.}\) It is now a straightforward exercise to show that \(\tau\vert_{D_2}\) satisfies the following properties:
\begin{align} \tau\vert_{D_2}(f)\amp= \tau\vert_{D_1}(f) \text{ for all } f\in P(W; x_0,x_0)\tag{2.15.7}\\ [f]=[g]\amp \implies \tau\vert_{D_2}(f)=\tau\vert_{D_2}(g) \text{ for all } f,g\in P(W)\tag{2.15.8}\\ \tau\vert_{D_2}(f*g) \amp =\tau\vert_{D_2}(f)\tau\vert_{D_2}(g) \text{ for all } f,g\in P(W)\text{,}\tag{2.15.9} \end{align}
where \(W=U\) or \(W=V\text{.}\) For example, (2.15.8) follows from the fact that if \(f\) and \(g\) are homotopic in \(W\text{,}\) then so are \(\alpha_x*f*\overline{\alpha}_y\) and \(\alpha_x*g*\overline{\alpha}_y\text{.}\) And (2.15.9) can be shown as follows: if \(f\in P(W; x,y)\) and \(g\in P(W; y,z)\text{,}\) then we have
\begin{align*} \tau\vert_{D_2}(f*g) \amp = \tau\vert_{D_1}(\alpha_x*f*g*\overline{\alpha}_z)\\ \amp =\tau\vert_{D_1}(\alpha_x*f*\overline{\alpha}_y*\alpha_y*g*\overline{\alpha}_z) \amp \knowl{./knowl/xref/eq_SvK_1a.html}{\text{(2.15.5)}}\\ \amp=\tau\vert_{D_1}(\alpha_x*f*\overline{\alpha}_y)\tau\vert_{D_1}(\alpha_y*g*\overline{\alpha}_z) \amp \knowl{./knowl/xref/eq_SvK_2a.html}{\text{(2.15.6)}}\\ \amp =\tau\vert_{D_2}(f)\tau\vert_{D_2}(g) \end{align*}

Construction: \(\tau\text{,}\) \(D=P(X)\).

We now define \(\tau\colon P(X)\rightarrow H\text{.}\) Given \(f\in P(X)\text{,}\) since \(f^{-1}(U)\cup f^{-1}(V)\) is an open cover of the compact set \(I=[0,1]\text{,}\) by Corollary 1.15.8 we can find a partition
\begin{equation*} 0=s_0< s_1 < \cdots < s_n=1 \end{equation*}
such that for all \(1\leq k\leq n\) we have \(f([s_{k-1},s_k])\subseteq U\) or \(f([s_{k-1},s_k])\subseteq V\text{.}\) Let \(\phi_k\colon I\rightarrow [s_{k-1}, s_k]\) be the “positive linear map” from \(I\) to \([s_{k-1}, s_k]\) (i.e., \(\phi_k(t)=(s_k-s_{k-1})t+s_{k-1}\)), and let \(f_k=f\circ \phi_k\text{.}\) We define
\begin{equation} \tau(f)=\prod_{k=1}^n\tau\vert_{D_2}(f_k)\text{.}\tag{2.15.10} \end{equation}
We must show that \(\tau\) is well-defined, agrees with \(\tau\vert_{D_2}\) and satisfies (2.15.1)–(2.15.2).

\(\tau\) is well-defined and agrees with \(\tau\vert_{D_2}\).

First we show that \(\tau\) is well-defined: i.e., that our definition does not depend on our choice of partition of \(I\text{.}\) Since any two partitions have a common refinement, it suffices to show that refining our partition by one additional endpoint does not affect our definition of \(\tau(f)\text{.}\) Suppose we split the \(k\)-th subinterval as
\begin{equation*} s_{k-1}< s_* < s_k\text{,} \end{equation*}
and let \(f_{k'}=f\circ \phi'\text{,}\) where \(\phi'\) is the positive linear map from \(I\) to \([s_{k-1},s_*]\text{,}\) and let \(f_{k''}=f\circ \phi''\text{,}\) where \(\phi''\) is the positive linear map from \(I\) to \([s_{*},s_k]\text{.}\) Since \(f_{k'}, f_{k''}\in P(W)\) for \(W=U\) or \(W=V\text{,}\) we have
\begin{align*} \tau(f)\amp= \prod_{l=1}^{k-1}\tau\vert_{D_2}(f_l)\cdot\left(\tau\vert_{D_2}(f_{k'})\cdot\tau\vert_{D_2}(f_{k''})\right)\cdot \prod_{l=k+1}^n\tau\vert_{D_2}(f_{l}) \\ \amp= \prod_{l=1}^{k-1}\tau\vert_{D_2}(f_l)\cdot\tau\vert_{D_2}(f_{k'}*f_{k''})\cdot \prod_{l=k+1}^{n}\tau\vert_{D_2}(f_l)\\ \amp = \prod_{l=1}^{n}\tau\vert_{D_2}(f_l)\text{,} \end{align*}
where the last equality holds because \([f_k]=[f_{k'}*f_{k''}]\) and property (2.15.8).
It is clear that \(\tau\) agrees with \(\tau\vert_{D_2}\) since for a path \(f\in P(W)\text{,}\) where \(W=U\) or \(W=V\text{,}\) we may choose as our partition
\begin{equation*} s_0=0< s_1=1\text{,} \end{equation*}
in which case \(\tau(f)=\tau\vert_{D_{2}}(f_1)=\tau\vert_{D_2}(f)\text{.}\)
In particular since \(\tau\) agrees with our earlier definition of \(\tau\vert_{D_1}\text{,}\) where \(D_1=P(U; x_0,x_0)\cup P(V; x_0,x_0)\text{,}\) properties (2.15.3)–(2.15.4) are automatically satisfied.

\(\tau\) satisfies (2.15.1).

Assume \(f,g\in P(X; x,y)\) and \([f]=[g]\text{.}\) Let \(F\colon I\times I\rightarrow X\) be a path homotopy from \(f\) to \(g\text{.}\) Again, using Corollary 1.15.8, we can find partitions
\begin{align*} 0=s_0 \amp < s_1 < \dots < s_n=1 \\ 0=t_0 \amp < t_1 < \dots < t_n=1 \end{align*}
such that for each subrectangle \(R_{k,l}=[s_{k-1}, s_k]\times [t_{l-1},t_l]\) of \(I\times I\text{,}\) we have \(F(R_{k,l})\subseteq W\text{,}\) where \(W=U\) or \(W=V\text{.}\) For each \((k,l)\) with \(1 \leq k\leq n\) and \(0\leq l\leq n\text{,}\) let \(h_{k,l}\) be the horizontal path in \(I\times I\) from \((s_{k-1}, t_l)\) to \((s_k, t_l)\text{;}\) similarly define \(v_{k,l}\) to be the vertical path from \((s_{k}, t_{l-1})\) to \((s_k,t_l)\) for all pairs \((k,l)\) with \(0\leq k\leq n\) and \(1\leq l\leq n\text{.}\) Lastly, define
\begin{align*} f_{k,l} \amp =F\circ h_{k,l}\\ \gamma_{k,l} \amp =F\circ v_{k,l} \end{align*}
for all pairs \((k,l)\) for which these expressions make sense. Fix a pair \((k,l)\) with \(1\leq k,l\leq n\text{.}\) The polygonal paths \(h_{k,l-1}*v_{k,l}\) and \(v_{k-1,l}*h_{k,l}\) respectively traverse the bottom-and-right segment and left-and-top segment of the rectangle \(R_{k,l}\text{.}\) Since the rectangle is convex, these paths are homotopic in \(R_{k,l}\text{.}\) Composing with \(F\) gives us a homotopy of \(f_{k,l-1}*\gamma_{k,l}\) and \(\gamma_{k-1,l}*f_{k,l}\) in \(F(R_{k,l})\subseteq W\text{.}\) Using properties of \(\tau\vert_{D_2}\text{,}\) we conclude that
\begin{equation*} \tau\vert_{D_2}(f_{k,l-1})\tau\vert_{D_2}(\gamma_{k,l})=\tau\vert_{D_2}(\gamma_{k-1,l})\tau\vert_{D_2}(f_{k,l}) \end{equation*}
and hence that
\begin{equation} \tau\vert_{D_2}(f_{k,l-1})=\tau\vert_{D_2}(\gamma_{k-1,l})\tau\vert_{D_2}(f_{k,l})\tau\vert_{D_2}(\gamma_{k,l})^{-1}\text{.}\tag{2.15.11} \end{equation}
Let \(f_{l}(s)=F(s,t_l)\) for all \(0\leq j\leq n\text{.}\) So in particular, we have \(f_{0}(s)=F(s,0)=f(s)\) and \(f_n(s)=F(s,1)=g(s)\text{.}\) Note that we have
\begin{equation*} f_j\circ \phi_k=F\circ h_{k,l}=f_{k,l}\text{.} \end{equation*}
It follows, using our definition of \(\tau\) that for all \(1\leq l\leq n\text{,}\) we have
\begin{align*} \tau(f_{l-1}) \amp =\prod_{k=1}^n\tau\vert_{D_2}(f_{k,l-1})\\ \amp = \prod_{k=1}^n(\tau\vert_{D_2}(\gamma_{k-1,l})\tau\vert_{D_2}(f_{k,l})\tau\vert_{D_2}(\gamma_{k,l})^{-1}) \amp \knowl{./knowl/xref/eq_conj.html}{\text{(2.15.11)}}\\ \amp = \tau\vert_{D_2}(e_{x})\cdot \prod_{k=1}^n\tau\vert_{D_2}(f_{k,l})\cdot \tau\vert_{D_2}(e_y) \amp \text{(cancellation)}\\ \amp = \prod_{k=1}^n\tau\vert_{D_2}(f_{k,l}) \amp (\tau\vert_{D_2}(e_x)=\tau\vert_{D_2}(e_y)=e)\\ \amp = \tau(f_l)\text{.} \end{align*}
We conclude that \(\tau(f)=\tau(f_{0})=\tau(f_{n})=\tau(g)\text{,}\) as desired.

\(\tau\) satisfies (2.15.2).

Let \(f\in P(X; x,y)\) and \(g\in P(X; y,z)\text{.}\) We prove that \(\tau(f*g)=\tau(f)\tau(g)\text{.}\) We choose a partition
\begin{equation*} 0=s_0< s_1 < \dots < s_n=1 \end{equation*}
such that \(s_k=1/2\) for some \(1\leq k\leq n\) and for all \(1\leq l\leq n\) we have \(f([s_{l-1}, s_l])\subseteq W\text{,}\) where \(W=U\) or \(W=V\text{.}\) As above, we let \(\phi_l\colon I\rightarrow [s_{l-1}, s_l]\) be the positive linear map from \(I\) onto \([s_{l-1}, s_l]\text{.}\) First observe, using the definition of \(*\text{,}\) that
\begin{align*} f*g\circ\phi_l(s) \amp= \begin{cases} f(2\phi_l(s)) \amp \text{if } l\leq k\\ g(2\phi_l(s)-1) \amp \text{if } l\geq k+1 \end{cases} \end{align*}
By definition, we have
\begin{align*} \tau(f*g) \amp=\prod_{l=1}^n\tau\vert_{D_2}(f*g\circ \phi_l) \\ \amp = \prod_{l=1}^k\tau\vert_{D_2}(f\circ (2\phi_l))\cdot \prod_{l=k+1}^n\tau\vert_{D_2}(g\circ (2\phi_l-1)) \amp (\text{def. of } *) \\ \amp = \tau(f)\tau(g) \text{.} \end{align*}
The last equality here follows as the functions
\begin{align*} 2\phi_l\colon I\amp \rightarrow [2s_{l-1},2s_l] \amp (l\leq k)\\ (2\phi_l-1)\colon I\amp \rightarrow [2s_{l-1}-1,2s_l-1] \amp (l\geq k+1) \end{align*}
are the positive linear maps corresponding to the two partitions
\begin{align*} 0=2s_0 \amp < 2s_{1} < \dots < 2s_{k}=1 \\ 0=2s_{k}-1 \amp < 2s_{k+1}-1 < \dots < 2s_{n}-1=1 \text{.} \end{align*}
The definition of \(\tau\) now implies
\begin{align*} \tau(f) \amp =\prod_{l=1}^k\tau\vert_{D_2}(f\circ (2\phi_l))\\ \tau(g) \amp =\prod_{l=k+1}^n\tau\vert_{D_2}(g\circ (2\phi_l-1))\text{.} \end{align*}

Proof.

Proof.

This follows directly from the explicit description of \(\pi_1(X,x_0)\) given in Corollary 2.15.4.
PrevTopNext