A local to global argument on low dimensional manifolds
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A local to global argument on low dimensional manifolds. / Nariman, Sam.
I: Transactions of the American Mathematical Society, Bind 373, Nr. 2, 2020, s. 1307-1342.Publikation: Bidrag til tidsskrift › Tidsskriftartikel › Forskning › fagfællebedømt
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TY - JOUR
T1 - A local to global argument on low dimensional manifolds
AU - Nariman, Sam
N1 - Publisher Copyright: © 2019 American Mathematical Society.
PY - 2020
Y1 - 2020
N2 - 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.
AB - 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.
U2 - 10.1090/tran/7970
DO - 10.1090/tran/7970
M3 - Journal article
AN - SCOPUS:85080879865
VL - 373
SP - 1307
EP - 1342
JO - Transactions of the American Mathematical Society
JF - Transactions of the American Mathematical Society
SN - 0002-9947
IS - 2
ER -
ID: 270425035