A local to global argument on low dimensional manifolds

2017/06/14 by Nariman, Sam · 1 citation
#57R19 #57R32 #57R40 #57R50 #57R52 #57R65 #Algebraic Topology (math.AT) #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.1706.04602

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 BHomeo^δ(M)→ BHomeo(M) induces a homology isomorphism where Homeo^δ(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.

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