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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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A local to global argument on low dimensional manifolds
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by Sam Nariman PDF
Trans. Amer. Math. Soc. 373 (2020), 1307-1342 Request permission

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.
  • © Copyright 2019 American Mathematical Society
  • 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
  • MathSciNet review: 4068265