Return to Answer

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.

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.

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 $> 3$, 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.

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.

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 > 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 repesenting 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 of each component of each level set maps a subgroup of small rank under the inclusion into pi1(K). with( G) is defined as a Minmax over all slicings of all complexes K with pi1 K = G of the rank of these image subgroups. I wrote a few pages to show that width( Z^k) = k-1. The only slightly techichal ingrediaent is Lusternick-Schnirelmann category. This answers negatively these finiteness questoins since there are d manifolds with pi1 =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.

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 $> 3$, 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.