Fix a dimension $n\geqslant 2$. Let $S= \{M_1,\ldots, M_k\}$ be a finite set of smooth compact $n$-manifold with boundary. Let us say that a smooth closed $n$-manifold is generated by $S$ if it may be obtained by gluing some copies of elements in $S$ via some arbitrary diffeomorphisms of their boundaries.

For instance:

• Every closed orientable surface is generated by a set of two objects: a disc and a pair-of-pants $P$,
• Waldhausen's graph manifolds are the 3-manifolds generated by $D^2\times S^1$ and $P\times S^1$,
• The 3-manifolds having Heegaard genus $g$ are those generated by the handlebody of genus $g$ alone,
• The exotic $n$-spheres with $n\geqslant 5$ are the manifolds generated by $D^n$ alone.

A natural question is the following:

Fix $n\geqslant 3$. Is there a finite set of compact smooth $n$-manifolds which generate all closed smooth $n$-manifolds?

I expect the answer to be ''no'', although I don't see an immediate proof. In particular, I expect some negative answers to both of these questions:

Is there a finite set of compact 3-manifolds which generate all hyperbolic 3-manifolds?

and

Is there a finite set of compact 4-manifolds which generate all simply connected 4-manifolds?