## 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.## References

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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