In the article "A Division Theorem for Varieties" by S. E. Cappell,

https://www.maths.ed.ac.uk/~v1ranick/papers/capsplit.pdf

the following "inverse" of Seifert-van Kampen's theorem for closed varieties is indicated in the introduction (see page 71).

Let $ Y $ to be a connected connected dimension variety (say, differentiable) $> 4 $. Suppose that the fundamental group of $ Y $ is written as a merged product $ pi_ {1} (Y) = G_ {1} ast_ {H} G_ {2} $ two groups $ G_ {1} $ and $ G_ {2} $ along a common subgroup $ H leq G_ {1} cap G_ {2} $. Then there is a closed connected codimension $ 1 $ submanifold $ X subset Y $ such as $ Y setminus X $ has two components, let's say with closures $ Y_ {1} $ and $ Y_ {2} $ in $ Y $, such as $ pi_ {1} (X) = H $ and $ pi_ {1} (Y_ {j}) = G_ {j} $, $ j = 1, $ 2.

**My question:** How can this result be proven?

Cappell states that the evidence is easily derived from the methods developed in section I ยง3 of the document. However, I do not see how these methods can be adapted. In fact, the methods are based on the existence of a homotopy equivalence $ f colon Y rightarrow Y $ to another variety already divided by a submultiple of codimension $ X $, and we consider $ X = f ^ {- 1} (X) $ and wants to make the restriction of $ f $ at $ X rightarrow X $ an equivalence of homotopy by deformation $ f $ (and changing $ X $). In my question, however, it is not clear how to choose a first $ X $ work with. In addition, the sleeve exchange technique only seems useful to make the card induced $ pi_ {1} (X) rightarrow pi_ {1} (Y) $ injective (by killing elements in the kernel), but not to produce the desired groups $ pi_ {1} (X) = H $ and $ pi_ {1} (Y_ {j}) = G_ {j} $, $ j = 1, $ 2.