For a $n$-manifold $M$ with nonempty boundary $\partial M$, a collar neighborhood of $\partial M$ in $M$ is an open neighborhood of $M$ homeomorphic to $\partial M \times [0,1)$ by a homeomorphism taking $\partial M$ to $\partial M \times 0$.
It is proved in Proposition 3.42 of Hatcher's Algebraic Topology, that for a compact manifold $M$ with boundary, there is a collar neighborhood of $\partial M$.
Now, suppose $M$ is a compact $n$-manifold whose boundary $\partial M$ is decomposed as the union of two compact $(n-1)$-manifolds $A$ and $B$ with a common boundary $\partial A=\partial B=A\cap B$. Hatcher says that the existence of collar neighborhoods of $A \cap B$ in $A$ and $B$ and $\partial M$ in $M$ implies that $A$ and $B$ are deformation retracts of open neighborhoods $U$ and $V$ in $M$ such that $U \cup V$ deformation retracts onto $A\cup B=\partial M$ and $U\cap V$ deformation retracts onto $A\cap B$.
How does this hold? I don't see how do I have to choose $U$ and $V$.
Thanks in advance
Let $C$ be a collar neighborhood of $\partial M$ in $M$ and let $D$ and $E$ be collar neighborhoods of $A\cap B$ in $A$ and $B$, respectively. Note that $A\cup E$ then deformation retracts to $A$, by deformation retracting $E$ down to $A\cap B$, and similarly $B\cup D$ deformation retracts to $B$, and $D\cup E$ deformation retracts to $A\cap B$.
Identifying $C$ with $\partial M\times [0,1)$, let $U=(A\cup E)\times [0,1)$ and $V=(B\cup D)\times [0,1)$. Then $U$ deformation retracts onto $A\times\{0\}$ and $V$ deformation retracts to $B\times\{0\}$. Moreover $U\cup V=C$ deformation retrafts to $\partial M$ and $U\cap V=(D\cup E)\times[0,1)$ deformation retracts to $(A\cap B)\times\{0\}$. So, $U$ and $V$ have all the desired properties.