2020/07/01 by Gerrit Herrmann, Herrmann, Gerrit, José Pedro Quintanilha +1
Mathematics · Computer Science · #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2007.00522
For a compact oriented smooth n-manifold M and a codimension-1 homology class ϕ∈ Hn-1(M, ∂ M), we investigate a simplicial complex S^†(M, ϕ) relating the properly embedded hypersurfaces in M representing ϕ. 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^†(M, ϕ) is connected and simply connected in every dimension n. We also show connectedness of a similar complex T^†(M, ϕ) adapted to the 3-dimensional case, where only Thurston norm-realizing surfaces are considered. The connectedness results are transported to the complexes S(M, ϕ), T(M, ϕ) 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 2-complexes are used to represent 1-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 connectedness of T^†(M, ϕ) to define a new ℓ2-invariant of 2-dimensional homology classes in irreducible and boundary-irreducible oriented compact connected 3-manifolds with empty or toroidal boundary.