Three slices of 3-manifold topology: spatial graphs, surface complexes and simplicial volumes

This thesis presents three research projects, with one chapter devoted to each. The first chapter establishes a canonical decomposition of piecewise-linear spatial graphs (that is, graphs embedded in 3-spheres), into pieces that are non-separable by 2-spheres, and have no topologically cut vertices -- such spatial graphs are called "blocks". This was motivated by the goal of producing an algorithm for testing the isomorphism type of spatial graphs, by encoding them as 3-manifolds with boundary pattern and applying a theorem of Matveev. Operations of disjoint union and vertex sum of spatial graphs are defined, and a combinatorial gadget for specifying iterated vertex sums is introduced, called a "tree of spatial graphs". The main results of this chapter are that each spatial graph is the disjoint union of non-separable pieces in a unique way, and each piece is the realization of a unique tree of blocks. Chapter 2 studies a simplicial complex defined for compact oriented smooth manifolds M (of any dimension) with a chosen codimension-1 homology class φ. This complex S*(M, φ) relates the properly embedded hypersurfaces in M representing φ, and its definition is similar 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 show that S*(M, φ) is connected and simply connected. We also show connectedness of a similar complex adapted to the 3-dimensional case, where only Thurston norm-realizing surfaces are considered. The connectedness results are transported to the complexes where hypersurfaces are taken up to isotopy, and in dimension 2 also the simple connectedness result is extended. The last chapter presents the proof that oriented compact connected 3-manifolds with toroidal boundary that are prime and not covered by S^3 satisfy integral approximation of simplicial volume. The computation of stable integral simplicial volume for such a manifold M is carried out in terms of its integral foliated simplicial volume, which in turn is obtained by assembling the values for each of the JSJ pieces of M. The assembly requires a variant of integral foliated simplicial volume that takes into account the norm of the boundary of fundamental cycles. The aim of this chapter is to highlight the overall strategy of the argument, breaking down the main ingredients of the proof, so some of the more technically involved arguments are omitted