A local to global argument on low dimensional manifolds
A local to global argument on low dimensional manifolds
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DOI：
10.1090/tran/7970
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发表时间：
2017-06
影响因子：
1.3
通讯作者：
Sam Nariman
中科院分区：
文献类型：
--
作者：
Sam Nariman
对于维数小于4 4的定向流形M M，我们利用与其子流形相关联的某些复形的可收缩性，将M M切割成更简单的片段，以便进行局部到全局的论证.特别是，在这些方面，我们给出了叶理理论中Thurston的一个深层定理的另一种证明，即分类空间B Homeo δ ε（M）→ B Homeo ε（M）\mathrm {B}\operatorname {Homeo}^{\delta }（M）\到\mathrm {B} \operatorname {Homeo}（M）之间的自然映射诱导一个同调同构，其中Homeo δ ε（M）\operatorname {Homeo}^{\δ}（M）表示M的离散同胚群。我们的证明表明，在低维，瑟斯顿定理可以证明，而不使用叶理理论。最后，我们表明这种技术为低维同胚群的同伦类型提供了新的视角。特别地，我们给出了Hacher定理的一个不同的证明，即带边界的Haken 3 -流形的同胚群是同伦离散的，而不使用他的析取技巧。
For an oriented manifold M M whose dimension is less than 4 4 , we use the contractibility of certain complexes associated to its submanifolds to cut M 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 B Homeo δ ( M ) → B Homeo ( M ) \mathrm {B}\operatorname {Homeo}^{\delta }(M)\to \mathrm {B} \operatorname {Homeo}(M) induces a homology isomorphism where Homeo δ ( M ) \operatorname {Homeo}^{\delta }(M) denotes the group of homeomorphisms of M 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 3 -manifolds with boundary are homotopically discrete without using his disjunction techniques.