Question

Thanks to Ian Agol for pointing out this question and a related one on levels of Morse functions - /305067/. In both case, for smooth manifold of dim $> 33$, 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 repesenting 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 pi1 $\pi_1$ of each component of each level set maps a subgroup of small rank under the inclusion into pi1(K). with$\pi_1(K)$. width( G) is defined as a Minmax over all slicings of all complexes K with pi1 $\pi_1 K = G G$ of the rank of these image subgroups. I wrote a few pages to show that width( Z^k$\mathbb{Z}^k $) $= k-1k-1$. The only slightly techichal ingrediaent technical ingredient is Lusternick-Schnirelmann category. This answers negatively these finiteness questoins questions since there are d $d$ manifolds with pi1 $\pi_1 =Z^k \mathbb{Z}^k$ all k, $k$, as long as d>3. $d>3$. As soon as the notes are teXed, I can post them on the arxiv or math overflow.