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