  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.
