In the paper "A splitting theorem for manifolds" by S.E. Cappell,
https://www.maths.ed.ac.uk/~v1ranick/papers/capsplit.pdf
the following "inverse" of the Seifert-van Kampen theorem for closed manifolds is stated in the introduction (see p. 71).
Let $Y$ be a closed connected (say, differentiable) manifold of dimension $> 4$. Suppose that the fundamental group of $Y$ is written as amalgamated product $\pi_{1}(Y) = G_{1} \ast_{H} G_{2}$ of two groups $G_{1}$ and $G_{2}$ along a common subgroup $H \leq G_{1} \cap G_{2}$. Then, there exists a closed connected codimension $1$ submanifold $X \subset Y$ such that $Y \setminus X$ has two components, say with closures $Y_{1}$ and $Y_{2}$ in $Y$, such that $\pi_{1}(X) = H$ and $\pi_{1}(Y_{j}) = G_{j}$, $j = 1, 2$.
My question: How can this result be proved?
Cappell claims that the proof follows easily from the methods developed in Section I §3 of the paper. However, I do not see how these methods there can be adapted. In fact, the methods rely on the existence of a homotopy equivalence $f \colon Y \rightarrow Y'$ to another manifold that is already split by a codimension submanifold $X'$, and one considers $X = f^{-1}(X')$ and wants to make the restriction of $f$ to $X \rightarrow X'$ a homotopy equivalence by deforming $f$ (and changing $X$). In my question, it is however not clear how to choose an initial $X$ to work with. Moreover, the handle exchanging technique seems only be useful for making the induced map $\pi_{1}(X) \rightarrow \pi_{1}(Y)$ injective (by killing elements in the kernel), but not for producing the desired groups $\pi_{1}(X) = H$ and $\pi_{1}(Y_{j}) = G_{j}$, $j = 1, 2$.
