A local to global argument on low dimensional manifolds

For an oriented manifold MM whose dimension is less than 44, we use the contractibility of certain complexes associated to its submanifolds to cut MM 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 BHomeoδ⁡(M)→BHomeo⁡(M)\mathrm {B}\operatorname {Homeo}^{\delta }(M)\to \mathrm {B} \operatorname {Homeo}(M) induces a homology isomorphism where Homeoδ⁡(M)\operatorname {Homeo}^{\delta }(M) denotes the group of homeomorphisms of MM 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 33-manifolds with boundary are homotopically discrete without using his disjunction techniques.