The Seifert–van Kampen Theorem: Examples

For a path $\eta$ from $w$ to $z$, define $c_{\eta }([\alpha ])=[\bar{\eta }\ast \alpha \ast \eta ]$. Endpoint-fixed homotopies are preserved by concatenating fixed paths, while the standard cancellation homotopies for $\eta \ast \bar{\eta }$ and $\bar{\eta }\ast \eta$ show that $c_{\eta }$ is a homomorphism with inverse $c_{\bar{\eta }}$. Thus both $\rho$ and $\sigma$ define basepoint-change isomorphisms.

Since $\sigma =b\ast \rho$, reversal of concatenation gives $c_{\sigma }([\alpha ])=[\bar{\rho }\ast \bar{b}\ast \alpha \ast b\ast \rho ]=c_{\rho }(b^{-1}[\alpha ]b)\mathrm{.}$ Hence $c_{\bar{\rho }}{c}_{\rho }(a)=a$, whereas $c_{\bar{\rho }}{c}_{\sigma }(a)=b^{-1}ab$ in the identification [L1].

The words $a$ and $b^{-1}ab$ are distinct reduced words by [F2]. Therefore their images under the isomorphism $c_{\rho }$ are distinct, so $c_{\rho }\mathrm{\ne }{c}_{\sigma }$.

Specializing [F1] to $n=2$ gives a retraction $r:P\to C$ and an endpoint-fixed homotopy from ${\mathrm{id}}_P$ to the composite of $r$ with the inclusion $i:C\hookrightarrow P$, fixing $(1,0)$.

Functoriality gives $r_{\ast }{i}_{\ast }=\mathrm{id}$ and the pointed homotopy in step 1.1 gives $i_{\ast }{r}_{\ast }=\mathrm{id}$, so $i_{\ast }$ is an isomorphism ${\pi }_1(C,(1,0))\cong {\pi }_1(P,(1,0))$.

The pointed homeomorphism of [F3] induces an isomorphism from the quotient-circle fundamental group to ${\pi }_1(C,(1,0))$. Composing it with [F4] and the isomorphism of step 2.1 gives ${\pi }_1(P,(1,0))\cong \mathbb{Z}$.

By [L1], stereographic projection is a pointed homeomorphism $S^2\setminus \{N\}\cong {\mathbb{R}}^2$ for the chosen basepoints.

The same stereographic projection restricts by [L1] to a pointed homeomorphism $S^2\setminus \{N,S\}\cong {\mathbb{R}}^2\setminus \{0\}$.

The plane is nonempty and convex, so [F1] makes its fundamental group trivial; [F2] transports that calculation through step 1.1.

By [L2] the latter space has fundamental group $\mathbb{Z}$, and [F2] transports this group through step 1.2.

Under [L1], the products $ab$ and $ba$ are represented by the two reduced words with syllable sequences $(a,b)$ and $(b,a)$.

These reduced words are distinct by [F1], so $ab\mathrm{\ne }ba$ in ${\pi }_1(W,w)$.

Thus the fundamental group of the two-circle wedge is not abelian, providing a counterexample to the universal claim.

Both defining intervals have length $3\mathrm{/}4<1$, so [F1] makes $U$ and $V$ open arcs. Their displayed lifts show $U\cup V=Q$, while $U\cap V=p((-1\mathrm{/}8,1\mathrm{/}8))\bigsqcup p((3\mathrm{/}8,5\mathrm{/}8)),$ a disjoint union of two nonempty open arcs. The basepoint lies in the first component $C=p((-1\mathrm{/}8,1\mathrm{/}8))$.

The three arcs $U,V,C$ are homeomorphic to open intervals, which are nonempty and convex. Hence [F2] and [F4] make all three fundamental groups trivial.

The pushout of the two homomorphisms from the trivial group to the two trivial factor groups is itself the trivial group: for every target group there is exactly one compatible pair of homomorphisms and exactly one homomorphism from the trivial group.

The actual group ${\pi }_1(Q,[0])$ is isomorphic to $\mathbb{Z}$ by [F3], so it is nontrivial and cannot be the pushout computed in step 3.1. Thus the false claim fails for this cover, and path-connectedness of the full overlap cannot be omitted from the two-set theorem.