We prove that there exists an algorithm for determining whether two piecewise-linear spatial graphs are isomorphic. In its most general form, our theorem applies to spatial graphs furnished with vertex colourings, edge colourings and/or edge orientations.

We first show that spatial graphs admit canonical decompositions into blocks, that is, spatial graphs that are non-split and have no cut vertices, in a suitable topological sense. Then, we apply a result of Haken and Matveev in order to algorithmically distinguish these blocks.

In this paper, we generalize the original idea of Thurston for the so-called Mather-Thurston’s theorem for foliated bundles to prove new variants of this theorem for PL homeomorphisms and contactormorphisms. These versions answer questions posed by Gelfand-Fuks ([GF73, Section 5]) and Greenberg ([Gre92]) on PL foliations and Rybicki ([Ryb10, Section 11]) on contactomorphisms. The interesting point about the original Thurston’s technique compared to the better-known Segal-McDuff’s proof of the Mather-Thurston theorem is that it gives a compactly supported c-principle theorem without knowing the relevant local statement on open balls. In the appendix, we show that Thurston’s fragmentation implies the non-abelian Poincare duality theorem and its generalization using blob complexes ([MW12, Theorem 7.3.1]).

To the memory of John Mather.

Le nombre chromatique relatif $c_0(S)$ d’une surface compacte S à bord est défini comme la borne supérieure des nombres chromatiques des graphes plongés dans S avec tous leurs sommets sur $\partial S$ . Cet invariant topologique a été introduit pour l’étude de la multiplicité de la première valeur propre de Steklov sur S. Dans cet article, on montre que $c_0(S)$ est aussi pertinent pour l’étude des plongements polyédraux tendus de S en établissant deux résultats. Le premier est que s’il existe un plongement polyédral tendu de S dans $\mathbb {R}^n$ qui n’est pas contenu dans un hyperplan, alors $n\leq c_0(S)-1$ . Le second est que cette inégalité est optimale pour les surfaces de petit genre.

We construct infinitely many compact, smooth 4-manifolds which are homotopy equivalent to $S^{2}$ but do not admit a spine (that is, a piecewise linear embedding of $S^{2}$ that realizes the homotopy equivalence). This is the remaining case in the existence problem for codimension-2 spines in simply connected manifolds. The obstruction comes from the Heegaard Floer $d$ invariants.

We classify the compact 3-manifolds whose boundary is a union of 2-spheres, and which embed in T ×I ×I, where T is a triod and I the unit interval. This class is described explicitly as the set of punctured handlebodies. We also show that any 3-manifold in T × I × I embeds in a punctured handlebody.