Categorical group invariants of 3-manifolds

Abstract

For a given class G of groups, a 3-manifold M is of G-category ≤ k if it can be covered by k open subsets such that for each path-component W of the subsets the image of its fundamental group π 1 (W ) → π(M) belongs to G. The smallest number k such that M admits such a covering is the G-category, cat G (M). If M is closed, it has G-category between 1 and 4. We characterize all closed 3-manifolds of G-category 1, 2, and 3 for various classes G.

Old and new categorical invariants of manifolds

“…This class G S can be described in purely algebraic ways and can be shown to be closed under subgroups [22]. Note that hunf ⊃ ame ⊃ solv ⊃ G S .…”

“…It can be shown (Lemma 1 of [22]) that if G is a nonempty class of groups closed under subgroups and G f is the class consisting of the finitely generated members of G, then, for any compact manifold M, cat…”

“…The techniques developed in [20,22,23] may be applied to obtain results for cat K (M) for other complexes K , by first considering the corresponding cat G K (M). For example, when K is a torus or Klein bottle, the method developed in [23] is used in [30] to obtain a list of possible closed 3-manifolds M with cat K (M) = 2.…”

“…Thus we conjecture that cat {cyclic} (M 3 ) ≤ 3 if and only if M 3 is a connected sum of Seifert bundles. Slightly bigger classes are G K = {cyclic groups, Z × Z, K , Z 2 * Z 2 } and G T = {cyclic groups, Z × Z}.We have hunf ⊃ ame ⊃ solv ⊃ G K ⊃ G T ⊃ {cyclic}, hunf ⊃ ame ⊃ solv ⊃ abel ⊃ G T ⊃ {cyclic} and obtain[22]: Theorem 8 Let M be a closed 3-manifold.…”

Models of simply-connected trivalent 2-dimensional stratifolds

“…For example if G is a non-empty family of groups that is closed under subgroups, one would like to determine which (closed) 3-manifolds have G-category equal to 3. In [5] it is shown that such manifolds have a decomposition into three compact 3-submanifolds H 1 , H 2 , H 3 , where the intersection of H i ∩ H j (for i = j) is a compact 2-manifold, and each H i is G-contractible (i.e. the image of the fundamental group of each connected component of H i in the fundamental group of the 3-manifold is in the family G).…”