Geometric Topology 21

The van Kampen Theorem

We shall state versions of this result without proof.

Recall that $\pi_1(X,x_0)$ denotes the fundamental group of a topological space $X$ consisting of equivalence classes of loops based at $x_0$ (under $\approxeq$). If $X$ is path-connected we saw previously that it is independent of $x_0$ (up to isomorphism).

The vK theorem will allow us to compute $\pi_1$ of a space $X$ which is the union of two open subsets $U,V$ with path-connected intersection $U \cap V$ (the sphere $S^2$ being a case in point). The inclusions $i,j,I,J$ give rise to induced homomorphisms $i_*,j_*,I_*,J_*$ by composition $i_* : [\alpha] \mapsto [i \circ \alpha]$ etc.

and

Where we have chosen a basepoint $x_0 \in U \cap V$ and $\pi_1(A)$ denotes $\pi_1(A,x_0)$. The resulting diagrams commute.

Homotopy of Maps and Spaces

This notion was introduced in a previous section. Let $X,Y$ be path-connected spaces.

Definitions:

1. Two maps $f_0,f_1 : X \rightarrow Y$ are homotopic (written $f_0 \simeq f_1$) if there exists a map $H: X \times I \rightarrow Y$ such that $H_0 = f_0$ and $H_1 = f_1$.
2. The spaces $X \times Y$ are themselves called homotopic (written $X \simeq Y$) if there exist maps $f: X \rightarrow Y$ and $g: Y \rightarrow X$ such that $g \circ f \simeq 1_X$ and $f \circ g \simeq 1_Y$.
3. If $X \subset Y$ then $X$ is called a (strong) deformation retract of $Y$ if $f$ is the inclusion, $g \circ f = 1_X$, and $f \circ g \simeq_H 1_Y$ via a homotopy $H$ such that $H_t(x) = x \ \forall x \in X$.

Example: $S^1$ is a deformation retract of $Y = \mathbb{C}^* = \mathbb{C} - \{0\}$. To see this, take $g(z) = \frac{z}{|z|}$ and $H(z,t) = \frac{(1-t)z}{|z|} + tz$.

Proposition: If $X \simeq Y$ then $\pi_1(X) \cong \pi_1(Y)$.

This is easy to prove in the case of a deformation retract, by composing loops with $g$. The proposition generalises a previous corollary.

First Version of van Kampen's Theorem

Theorem vK1: If $U \cap V$ is simply-connected, then $\pi_1(X) \cong \pi_1(U) {\Large*} \pi_1(V)$.

Example: In the description of a torus $T$ as the quotient of a square $\mathscr{P}$, its boundary $\partial \mathscr{P}$ maps to the 'wedge' $S^1 \vee S^1$ of two circles, homeomorphic to the figure-eight. Call this space $X$. The circles can be enlarged into open sets $U,V$ with $U,V \simeq S^1$, and $U \cap V$ a 'cross'. This cross is homotopic to a point: the arcs can be continuously shortened until the figure has been collapsed to the central vertex $v$. This point is a deformation retract of the cross; one says that $U \cap V$ retracts to $v$.

Then vk1 and the previous proposition imply that:

If we regard the figure-eight $X$ as a graph, then $\{v\}$ is itself a minimal spanning tree (MST). It is a general result that the fundamental group of a graph is isomorphic to $F_n$ where $n=E-(V-1)$ is the number of edges not in a MST.

Second Version of van Kampen's Theorem

If $K = \pi_1(U \cap V)$ is non-trivial, it will modify the free-product $\pi_1(U) {\Large*} \pi_1(V)$. The idea is that the classes of loops in $U \cap V$ (which $i_*$ and $j_*$ map to $\pi_1(U)$ and $\pi_1(V)$) cannot count twice in computing $\pi_1(U \cap V)$.

Theorem 2 [van Kampen, Seifert]: With $U,V$ still open and $U \cap V$ path-connected, the fundamental group of $X = U \cup V$ is isomorphic to the amalgamated product:

where $N$ is the smallest normal subgroup containing all elements $i_*(k)^{-1}j_*(k), k \in K$.

Example: Take $X = T$. Let $o$ be the centre of the square $\mathscr{P}$, $\hat{o}$ its image in $T = \hat{\mathscr{P}}$, and $U = T - \{\hat{o}\}$. Then $U$ retracts to the boundary $\hat{\partial \mathscr{P}}$ of the unique face of $T$, and:

Now take $V$ to a small open disk (or square) containing $\hat{o}$. Then $U \cap V$ is homotopic to $\partial \mathscr{P}$ and a circle, and $i_*$ maps the generator of $\pi_1(U \cap V)$ to the class $aba^{-1}b^{-1}$ in $\pi_1(U)$ whilst $\pi_1(V) = \{e\}$ is trivial. Therefore $N$ is the smallest normal subgroup of $\pi_1(U) {\Large*} \pi_1(V) \cong F_2$ containing $aba^{-1}b^{-1}$, and $\pi_1(T) \cong \langle a,b | aba^{-1}b^{-1} \rangle$.

The Fundamental Groups of Spheres

Theorem: $S^n$ is simply connected (i.e. $\pi_1(S^n) \cong \{e\}$) if $n \geq 2$.

One proof exploits the fact that $S^n$ minus a point is homeomorphic to $\mathbb{R}^n$. Any loop in $S^n$ can then be regarded as a loop in $\mathbb{R}^n$ based at the origin, and $H(s,t) = (1-t)\alpha(s)$ deforms it to the constant loop.

Proof using vK2: Fix $n \geq 2$. Then $S^n$ is the union of $3/4$-spheres:

each homeomorphic to a disk, and simply-connected. The 'equator' $x_{n+1} = 0$ can be identified with $S^{n-1}$, and is path-connected for $n \geq 2$. By theorem vK2, $\pi_1(S^n)$ is a quotient of the free group $F = \pi_1(U) {\Large*} \pi_1(V) = \{e\}$, and is therefore trivial. $\blacksquare$

If we accept that $S^2$ is simply-connected, we can deduce by induction that $S^n$ is for $n \geq 3$, using only vk1. For the equator $S^{n-1}$ above is a deformation retract of $U \cap V$. So now all of $U,V,U \cap V$ are simply-connected for $n \geq 3$.