Abstract
For a given class \({\mathcal{G}}\) of groups, a 3-manifold M is of \({\mathcal{G}}\)-category \({\leq k}\) 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 \({ \pi_1(W) \rightarrow \pi(M )}\) belongs to \({\mathcal{G}}\). The smallest number k such that M admits such a covering is the \({\mathcal{G}}\)-category, \({cat_{\mathcal{G}}(M)}\). If M is closed, it has \({\mathcal{G}}\)-category between 1 and 4. We characterize all closed 3-manifolds of \({\mathcal{G}}\)-category 1, 2, and 3 for various classes \({\mathcal{G}}\).
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Gómez-Larrañaga, J.C., González-Acuña, F. & Heil, W. Categorical group invariants of 3-manifolds. manuscripta math. 145, 433–448 (2014). https://doi.org/10.1007/s00229-014-0697-3
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DOI: https://doi.org/10.1007/s00229-014-0697-3