The Complex of Hypersurfaces in a Homology Class

For a compact oriented smooth $n$-manifold $M$ and a codimension-$1$ homology class $ϕ\in \operatorname{H}_{n-1}(M, \partial M)$, we investigate a simplicial complex $\mathcal{S}^\dagger(M, ϕ)$ 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 $\mathcal{S}^\dagger(M, ϕ)$ is connected and simply connected in every dimension $n$. We also show connectedness of a similar complex $\mathcal{T}^\dagger(M, ϕ)$ adapted to the $3$-dimensional case, where only Thurston norm-realizing surfaces are considered. The connectedness results are transported to the complexes $\mathcal{S}(M, ϕ), \mathcal{T}(M, ϕ)$ 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 $\mathcal{T}^\dagger(M, ϕ)$ to define a new $\ell^2$-invariant of $2$-dimensional homology classes in irreducible and boundary-irreducible oriented compact connected $3$-manifolds with empty or toroidal boundary.

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