For an oriented manifold
M
M
whose dimension is less than
4
4
, we use the contractibility of certain complexes associated to its submanifolds to cut
M
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 that says the natural map between classifying spaces
B
Homeo
δ
(
M
)
→
B
Homeo
(
M
)
\mathrm {B}\operatorname {Homeo}^{\delta }(M)\to \mathrm {B} \operatorname {Homeo}(M)
induces a homology isomorphism where
Homeo
δ
(
M
)
\operatorname {Homeo}^{\delta }(M)
denotes the group of homeomorphisms of
M
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
3
-manifolds with boundary are homotopically discrete without using his disjunction techniques.