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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

A local to global argument on low dimensional manifolds


Author: Sam Nariman
Journal: Trans. Amer. Math. Soc. 373 (2020), 1307-1342
MSC (2010): Primary 57R32, 57R40, 57R50, 57R52, 57R65
DOI: https://doi.org/10.1090/tran/7970
Published electronically: November 1, 2019
MathSciNet review: 4068265
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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 that says the natural map between classifying spaces $\mathrm {B}\operatorname {Homeo}^{\delta }(M)\to \mathrm {B} \operatorname {Homeo}(M)$ induces a homology isomorphism where $\operatorname {Homeo}^{\delta }(M)$ 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.


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Additional Information

Sam Nariman
Affiliation: Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, Illinois 60208; and University of Copenhagen, Universitetsparken 5, 2100 Copenhagen, Denmark
MR Author ID: 1228088
Email: sam@math.northwestern.edu, sam@math.ku.dk

Received by editor(s): April 23, 2019
Received by editor(s) in revised form: July 16, 2019
Published electronically: November 1, 2019
Additional Notes: The author was partially supported by NSF DMS-1810644.
Article copyright: © Copyright 2019 American Mathematical Society