A local to global argument on low dimensional manifolds
Research output: Contribution to journal › Journal article › Research › peer-review
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.
Original language | English |
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Journal | Transactions of the American Mathematical Society |
Volume | 373 |
Issue number | 2 |
Pages (from-to) | 1307-1342 |
ISSN | 0002-9947 |
DOIs | |
Publication status | Published - 2020 |
Bibliographical note
Publisher Copyright:
© 2019 American Mathematical Society.
Links
- https://arxiv.org/pdf/1706.04602.pdf
Accepted author manuscript
ID: 270425035