0.2. The Seifert-van Kampen Theorem 3
is a neighborhood of y and h : U R+
n
is a homeomorphism, then
the last coordinate of h(y) is zero for every such pair (U, h).
0.1.4. The interior and boundary of a ∂-manifold are well defined. Specif-
ically, if M and N are ∂-manifolds and h : M N is a topological
homeomorphism, then h(∂M) = ∂N and h(Int M) = Int N.
0.1.5. If M is an n-dimensional ∂-manifold, then ∂M is an (n 1)-
dimensional manifold (without boundary).
0.1.6. If M and N are ∂-manifolds, then M × N is a ∂-manifold and
∂(M × N) = (∂M × N) (M × ∂N).
0.2. The Seifert-van Kampen Theorem
We will assume familiarity with the fundamental group and the theory of
covering spaces. The Seifert-van Kampen Theorem relates the fundamental
group of the union of two spaces to the fundamental groups of the two
constituent pieces. The setting for the theorem posits the following data:
U1 and U2 are pathwise connected, open subsets of a space X such that
X = U1 U2 and U0 = U1 U2 are pathwise connected, x U0, φi :
π1(U0,x) π1(Ui,x), i {1, 2}, and ψi : π1(Ui,x) π1(X, x), i {0, 1, 2},
are the inclusion-induced homomorphisms.
Theorem 0.2.1 (Seifert-van Kampen). If H is a group and ρi : π1(Ui,x)
H are any homomorphisms for i = 0, 1, 2 such that the diagram
π1(U1,x)
ρ1
π1(U0,x)
φ1
ρ0
φ2
H
π1(U2,x)
ρ2
is commutative, then there exists a unique homomorphism σ : π1(X, x) H
such that σψi = ρi for i = 0, 1, 2; that is, σ renders the following diagram
commutative:
π1(U1,x)
ρ1
ψ1
π1(U0,x)
φ1
ψ0
φ2
π1(X, x)
σ
H
π1(U2,x)
ρ2
ψ2
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