  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.

Geometric Topology