Herrmann, Gerrit and Quintanilha, Jose Pedro (2025) The Complex of Hypersurfaces in a Homology Class. MICHIGAN MATHEMATICAL JOURNAL, 75 (2). pp. 227-266. ISSN 0026-2285, 1945-2365
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For a compact oriented smooth n-manifold M and a codimension-1 homology class is an element of Hn-1(M, aM), we investigate a simplicial complex S dagger(M, ) relating the properly embedded hyper-surfaces in M representing . Its definition is akin to that of other classical complexes, such as the curve complex of a surface or the Kakimizu complex of a knot, with the difference that hypersurfaces are not taken up to isotopy. We prove that S dagger(M, ) is connected and simply connected in every dimension n. We also show the connectedness of a similar complex T dagger(M, ) adapted to the three-dimensional case, where only Thurston norm-realizing surfaces are considered. The connectedness results are transported to the complexes S(M, ) and T(M, ) where hypersurfaces are taken up to isotopy, and for n = 2, the simple connectedness result carries over as well. We also briefly discuss extensions to a context studied by Turaev, where regular graphs in 2complexes are used to represent one-dimensional cohomology classes. We finish with two applications: we give an alternative proof of the fact that all Seifert surfaces for a fixed knot in a rational homology sphere are tube-equivalent, and we use the connectedness of T dagger(M, ) to define a new 2-invariant pound of two-dimensional homology classes in irreducible and boundary-irreducible oriented compact connected 3-manifolds with empty or toroidal boundary.
| Item Type: | Article |
|---|---|
| Subjects: | 500 Science > 510 Mathematics |
| Divisions: | Mathematics |
| Depositing User: | Dr. Gernot Deinzer |
| Date Deposited: | 23 Jun 2026 08:17 |
| Last Modified: | 23 Jun 2026 08:17 |
| URI: | https://pred.uni-regensburg.de/id/eprint/66207 |
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