0.2. The Seifert-van Kampen Theorem 3 is a neighborhood of y and h : U → Rn + 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 φ 2 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) ρ 2 ψ2

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