In the article "A division theorem for multiples" 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 collectors is set out in the introduction (see p. 71).

Leave $ Y $ be a closed variety of connected dimensions (say, differentiable) $> $ 4. Suppose the fundamental group of $ Y $ It is written as amalgamated product $ pi_ {1} (Y) = G_ {1} ast_ {H} G_ {$} of two groups $ G_ {1} $ Y $ G_ {$} $ along a common subgroup $ H leq G_ {1} cap G_ {2} $. So, there is a closed co-dimension connected $ 1 $ submanifold $ X subset Y $ such that $ Y setminus X $ It has two components, say with closures $ Y_ {1} $ Y $ Y_ {2} $ in $ Y $such that $ pi_ {1} (X) = H $ Y $ pi_ {1} (Y_ {j}) = G_ {j} $, $ j = 1, $ 2.

**My question:** How can you prove this result?

Cappell states that the proof is easily deduced from the methods developed in Section I §3 of the document. However, I don't see how these methods can be adapted there. In fact, the methods are based on the existence of a homotopic equivalence. $ f colon Y rightarrow Y & # 39; $ to another collector that is already divided by a codimensional sub-block $ X & # 39; $and one considers $ X = f <-1> (X & # 39;) $ and wants to make the restriction of $ f $ to $ X right arrow X & # 39; $ an equivalence of deformation homotopy $ f $ (and changing $ X $) In my question, however, it is not clear how to choose an initial $ X $ work with. In addition, the handle exchange technique only seems to be useful for making the induced map $ pi_ {1} (X) rightarrow pi_ {1} (Y) $ injecting (killing elements in the nucleus), but not to produce the desired groups $ pi_ {1} (X) = H $ Y $ pi_ {1} (Y_ {j}) = G_ {j} $, $ j = 1, $ 2.