A local to global argument on low dimensional manifolds

TL;DR

This paper presents a new low-dimensional approach to prove Thurston's theorem on the homology of homeomorphism classifying spaces, avoiding foliation theory, and offers fresh insights into the homotopy types of homeomorphism groups.

Contribution

It introduces a novel local-to-global method for low-dimensional manifolds that simplifies proofs of key theorems in foliation theory and the homotopy properties of homeomorphism groups.

Findings

Thurston's theorem proved without foliation theory in dimensions less than 4.

Homeomorphism groups of Haken 3-manifolds are homotopically discrete.

New perspective on the homotopy type of homeomorphism groups in low dimensions.

Abstract

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.