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/. 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.

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.

It is not entirely clear in the question whether a "block" is a manifold with smooth boundary, or perhaps a manifold that is allowed to have ridges or more complicated corners. Let's assume that the boundary has to be smooth and that the blocks have to be glued along connected boundary components, because at the other extreme you can make any PL manifold from copies of a simplex.

If so, then Ian Agol's comment explains everything in dimension 3. As he explained, it's Ian's theorem that there is a non-Haken manifold $M$ of Heegaard genus $g' \ge g$ for every $g$. And, it follows from work of Scharlemann-Thompson and Casson-Gordon that a Morse function on such an $M$ must have a level set with a connected component of genus $\ge g'$. (And equality is trivial because a Heegaard surface is always a fattest Morse level set.)

If you have your blocks, you can always arrange them as a collection of "cups", i.e., you can pick a relative Morse function which is $0$ on every boundary component and negative in the interior. Then you can glue the boundary components of the blocks in pairs with "caps", which are copies of $\Sigma \times I$ with increasing Morse functions that begin at $0$ on their boundaries. (Or, equivalently, you could have a bipartite collection of blocks.) Since you can reuse the same Morse function on each cap or cup of a given type, having finitely many types implies a global bound on the genus of a connected component of a level set.