A local to global argument on low dimensional manifolds

Publikation: Bidrag til tidsskriftTidsskriftartikelfagfællebedømt

  • Sam Nariman

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 that says the natural map between classifying spaces BHomeoδ(M) → BHomeo(M) induces a homology isomorphism where Homeoδ(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.

OriginalsprogEngelsk
TidsskriftTransactions of the American Mathematical Society
Vol/bind373
Udgave nummer2
Sider (fra-til)1307-1342
ISSN0002-9947
DOI
StatusUdgivet - 2020

Links

ID: 270425035