The Seifert--van Kampen theorem for CW complexes

We say that a subset $A \subset X$ is a neighborhood deformation retract if there is an open set $U$ containing $A$ and a deformation retraction of $U$ onto $A$ (a map $h : U \to A$ with $h|_A = \operatorname {id}|_A$ so that $h$ is homotopic to the identity, with the homotopy restricting to the constant homotopy on $A$). If $(A, B)$ is a pair of sets, we say the pair is a neighborhood deformation retract if there are open sets $U, V$ with $A \subset U$, $B \subset V$ so that $U$ deformation-retracts to $A$, $V$ deformation-retracts to $B$, and $U \cap V$ deformation-retracts to $A \cap B$.

Show that if $(A, B)$ is a neighborhood deformation retract of $(U, V)$, then $A, B, A \cap B$ are all path connected iff $U, V, U \cap V$ are all path connected.

Recall (Exercise 5.4.1) that a pair $(A,B)$ is a neighborhood deformation retract of $(U,V)$ if $U, V$ are open sets with $A \subset U$, $B \subset V$, $U$ deformation-retracts to $A$, $V$ deformation-retracts to $B$, and $U \cap V$ deformation-retracts to $A \cap B$. Show that if $A, B, A \cap B$ are path connected and $(A, B)$ is a neighborhood deformation retract of $(U, V)$, then if $x_0 \in A \cap B$, there is an isomorphism between $\pi_1(A, x_0) *_{\pi_1(A \cap B, x_0)} \pi_1(B, x_0)$ and $\pi_1(U, x_0) *_{\pi_1(U \cap V, x_0)} \pi_1(V, x_0)$.

Show that if $X = A \cup B$, where $A, B, A \cap B$ are path connected, $x_0 \in A \cap B$, and $(A, B)$ is a neighborhood deformation retract (Exercise 5.4.1), then there is an isomorphism $\pi_1(A, x_0) *_{\pi_1(A \cap B, x_0)} \pi_1(B, x_0) \simeq \pi_1(X, x_0)$.

Suppose $M$ is a path-connected surface which is the union of two path-connected surfaces with boundary $N_1, N_2$ which intersect along a circle $C = \partial N_1 = \partial N_2$. Suppose there is a neighborhood $H$ of $C$ so that $(H, H \cap N_1, H \cap N_2, H \cap C)$ is homeomorphic to $(S^1 \times (-1,1), S^1 \times (-1,0], S^1 \times [0,1), S^1 \times \left\{ 0\right\} )$.

Apply Exercise 5.4.5 to compute

Suppose $K$ is a two-dimensional CW complex and $L$ is a subcomplex of $K$, so that $K$ can be formed from $L$ by inductively attaching cells of dimensions $0,1,2$. Using an inductive argument, show that $L$ is a neighborhood deformation retract (Exercise 5.4.1).

Suppose that $L, M$ are subcomplexes of the two-dimensional CW complex $K$. Show that $(L, M)$ is a neighborhood deformation retract (Exercise 5.4.1). (This, together with Exercise 5.4.7, in fact holds without the two-dimensional hypothesis.)

Prove the following theorem (the Seifert--van Kampen theorem for CW complexes): suppose $L, M$ are path-connected subcomplexes of the CW complex $K$ so that $L \cap M$ is path connected with $x \in L \cap M$. Then $\pi_1(K, x) \simeq \pi_1(L, x) *_{\pi_1(L \cap M, x)} \pi_1(M, x)$.