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From the classification of closed surfaces, it follows that any closed surface can be obtained by gluing discs and pants along their boundaries. I wonder whether a similar statement holds in higher dimensions.

Let $n\geq 3$. Does there exist a finite collection $\mathcal{C}$ of $n$-manifolds with boundary such that any closed $n$-manifold can be obtained from manifolds of $\mathcal{C}$ by identifying connected components of their boundaries?

I think a negative answer is known, but I am looking for precise references.

PS: If needed, you can consider that the manifolds here are smooth.

PSS: I asked the same question on math.stackexange.

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    $\begingroup$ Well, any 3-manifold has a Heegaard decomposition, which expresses it as two handlebodies glued together along their boundaries. Handlebodies can in turn be obtained from balls by identifying discs in their boundaries. But the existence of Heegaard decompositions is quite weak -- nothing like as strong as the classification of surfaces. $\endgroup$
    – HJRW
    Commented Oct 7, 2016 at 19:43
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    $\begingroup$ @HJRW: you are identifying parts of the boundary of the "building blocks", while the OP asks for connected components. Otherwise one could simply use handles. $\endgroup$
    – Marco Golla
    Commented Oct 7, 2016 at 19:51
  • $\begingroup$ Sorry, you're quite right, I misread the question. $\endgroup$
    – HJRW
    Commented Oct 7, 2016 at 20:11
  • $\begingroup$ My guess, as before, is that in 3D one should be able to prove that such a collection does not exist by using non-Haken hyperbolic 3-dimensional manifolds of arbitrarily high Heegaard genus since in this case Scharlemann-Thompson width equals the genus. I think, it will follow that one cannot take $C$ consisting of finitely many handlebodies and compression bodies. I am not sure about more complicated manifolds. $\endgroup$
    – Moishe Kohan
    Commented Oct 8, 2016 at 17:44

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