The Complex of Hypersurfaces in a Homology Class

TL;DR

This paper studies the topological properties of complexes formed by hypersurfaces representing a homology class in manifolds, proving their connectivity and simple connectivity, and applying these results to knot theory and 3-manifold invariants.

Contribution

It introduces and analyzes complexes of hypersurfaces in manifolds, establishing their connectivity properties and applying these to knot and 3-manifold theory.

Findings

The complexes are connected and simply connected in all dimensions.

Connectedness of related complexes is established for hypersurfaces up to isotopy.

Applications include alternative proofs in knot theory and new invariants for 3-manifolds.

Abstract

For a compact oriented smooth $n$-manifold $M$ and a codimension-$1$ homology class $\phi \in \operatorname{H}_{n-1}(M, \partial M)$, we investigate a simplicial complex $\mathcal{S}^\dagger(M, \phi)$ relating the properly embedded hypersurfaces in $M$ representing $\phi$. 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 $\mathcal{S}^\dagger(M, \phi)$ is connected and simply connected in every dimension $n$. We also show connectedness of a similar complex $\mathcal{T}^\dagger(M, \phi)$ adapted to the $3$-dimensional case, where only Thurston norm-realizing surfaces are considered. The connectedness results are transported to the complexes $\mathcal{S}(M, \phi), \mathcal{T}(M, \phi)$ 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 $\mathcal{T}^\dagger(M, \phi)$ 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.