TITLE: Can all n-manifolds be obtained by gluing finitely many blocks?
QUESTION [26 upvotes]: 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?

REPLY [12 votes]: I posted a paper on the arXiv, Group Width which answers this question for manifolds of dim $>3$ with sufficiently complicated fundamental group (there will be no finite set of blocks). As Greg Kuperberg said, there are many interesting variations which remain open and are a nice challenge to technique, e.g. the case of simply connected manifolds.