Simplicial Structures in Topology
Simplicial Structures in Topology
Simplicial Structures in Topology
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204 VI Homotopy Groups<br />
(VI.1.13) Theorem (Seifert–Van Kampen). Let |L| and |M| be two path-connected<br />
polyhedra such that |L ∩ M| is path-connected and not empty. Let a0 be a vertex<br />
shared by both polyhedra. The diagram<br />
π(|L ∩ M|,a0) π(iL)<br />
π(iM)<br />
��<br />
π(|M|,a0)<br />
π(iL)<br />
��<br />
π(|L|,a0)<br />
π(iM)<br />
��<br />
��<br />
π(|L ∪ M|,a0)<br />
(where iL and iM are the <strong>in</strong>clusion maps) is a pushout.<br />
Proof. We extend a spann<strong>in</strong>g tree |A(L∩M)| of |L∩M| to the spann<strong>in</strong>g trees |A(L)|<br />
and |A(M)| of |L| and |M|, respectively; the union<br />
|A(L)|∪|A(M)| = |A(L ∪ M)|<br />
is a spann<strong>in</strong>g tree of |L ∪ M| and so, by Lemma (V.2.6), it conta<strong>in</strong>s all vertices<br />
of L ∪ M.<br />
We now order the vertices of L∪M; this automatically gives the vertices of L and<br />
of M an order. By Theorem (VI.1.1), we know that π(|L∪M|,a0) is generated by the<br />
elements gij correspond<strong>in</strong>g to the ordered 1-simplexes {ai,aj} of L∪M �A(L∪M),<br />
with the relations gijg jkgki relative to the ordered 2-simplexes {ai,a j,ak} of L ∪ M.<br />
On the other hand, the pushout of groups π(|L|,a0) and π(|M|,a0) relative to the<br />
homomorphisms<br />
π(iL): π(|L ∩ M|,a0) → π(|L|,a0),<br />
π(iM): π(|L ∪ M|,a0) → π(|M|,a0),<br />
is determ<strong>in</strong>ed by the generators gij and hij correspond<strong>in</strong>g, respectively, to the<br />
ordered 1-simplexes of L � A(L) and M � A(M), with the relations gijg jkgki,<br />
hijh jkhki and<br />
π(iL)gij(π(iM)h ji) −1<br />
whenever gij = hij <strong>in</strong> L ∩ M.<br />
It is now easy to realize that the group π(|L∪M|,a0) described as above <strong>in</strong> terms<br />
of generators and relations co<strong>in</strong>cides with the pushout <strong>in</strong> question. �<br />
(VI.1.14) Corollary. If π(|L ∩ M|,a0)=0,thenπ(|L ∪ M|,a0) is the free product<br />
π(|L|,a0) ∗ π(|M|,a0).<br />
Let us consider a few th<strong>in</strong>gs before we turn to another result. Let |K| be a connected<br />
polyhedron and let α be a closed path, based at a vertex a0 ∈ K, that may be<br />
represented by the sequence of 1-simplexes:<br />
α = {a0,a1}.{a1,a2}.....{an,a0}.