A local to global argument on low dimensional manifolds

For an oriented manifold $M$ whose dimension is less than $4$, we use the contractibility of certain complexes associated to its submanifolds to cut $M$ into simpler pieces in order to do local to global arguments. In particular, in these dimensions, we give a different proof of a deep theorem of Thurston in foliation theory which says that the natural map between classifying spaces $\mathrm{B}\text{Homeo}^{\delta}(M)\to \mathrm{B}\text{Homeo}(M)$ induces a homology isomorphism where $\text{Homeo}^{\delta}(M)$ denotes the group of homeomorphisms of $M$ made discrete. Our proof shows that in low dimensions, Thurston's theorem can be proved without using foliation theory. Finally, we show that this technique gives a new perspective on the homotopy type of homeomorphism groups in low dimensions. In particular, we give a different proof of Hacher's theorem that the homeomorphism groups of Haken $3$-manifolds with boundary are homotopically discrete without using his disjunction techniques.