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:



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



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



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> Is there a finite set of compact 4-manifolds which generate all simply connected 4-manifolds?
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Can all n-manifolds be obtained by gluing finitely many blocks?
Thanks to Ian Agol for pointing out this question and a related one on levels of Morse functions - [/30567/](https://mathoverflow.net/questions/30567/). In both case, for smooth manifold of dim $> 3$, as expected, there is no finite list of blocks ( or regular level components.) The idea is that one may define the "width" of a group, by representing the group G as the fundamental group of some complex K and then slicing K into "levels". The game to to arrange the slices so that the image of $\pi\_1$ of each component of each level set maps a subgroup of small rank under the inclusion into $\pi\_1(K)$. width( G) is defined as a Minmax over all slicings of all complexes K with $\pi\_1 K = G$ of the rank of these image subgroups. I wrote a few pages to show that width( $\mathbb{Z}^k $) $= k-1$. The only slightly technical ingredient is Lusternick-Schnirelmann category. This answers negatively these finiteness questions since there are $d$ manifolds with $\pi\_1 =\mathbb{Z}^k$ all $k$, as long as $d>3$. As soon as the notes are teXed, I can post them on the arxiv or math overflow.


