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edited Apr 13, 2017 at 12:58

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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 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.

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.

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.

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created Oct 18, 2010 at 19:52

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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.