On the Diffeomorphism Group of S 1 S 2
On the Diffeomorphism Group of S 1 S 2
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关于S 1 S 2 的微分同胚群
DOI：
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发表时间：
1981
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影响因子：
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通讯作者：
A. Hatcher
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文献类型：
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作者：
A. Hatcher
本文的主要结果是微分同态群Diff(S×S)具有人们所期望的同伦型，即它的微分同态子群的同伦型，即用S的等距群O(3)的一个元素使每个球面{x}×S到一个球面{y}×S，其中函数x，y是O(2)的元素S的等距。不难看出，这个子群与乘积O(2)×O(3)×ΩSO(3)同胚，最后一个因子是SO(3)中基于恒等式的光滑环空间。这与连续环空间具有相同的同伦类型。这些环空间对所有i都有H_2i(ΩSO(3)；Z)非零，因此我们得出结论：DIFF(S×S)不是同伦等价于一个李群。曲面的微分同态群和许多不可约的3-流形与李群(实际上通常是离散群)是同伦等价的，而S×S是最简单的不是这样的流形。通过[H]中证明的斯梅尔猜想，同伦型差(S×S)的计算很容易归结为使S×S中两个球面的族不交的问题。更一般地说，设M是一个连通的三维流形，其中包含一个不约束球的球面S⊂M，并设S×[−1，1]⊂M是这位S的双叶邻域。设E是f：s2→M3的光滑嵌入空间，它的像不有球，E‘是f的子空间，它存在x∈[−1，1]，且f(S)与{x}×S不交.我们需要的结果是：
The main result of this paper is that the group Diff(S×S) of diffeomorphisms S1×S2→S1×S2 has the homotopy type one would expect, namely the homotopy type of its subgroup of diffeomorphisms that take each sphere {x}×S to a sphere {y}×S by an element of the isometry group O(3) of S , where the function x,y is an isometry of S , an element of O(2) . It is not hard to see that this subgroup is homeomorphic to the product O(2)×O(3)×ΩSO(3) , this last factor being the space of smooth loops in SO(3) based at the identity. This has the same homotopy type as the space of continuous loops. These loopspaces have H2i(ΩSO(3);Z) nonzero for all i , so we conclude that Diff(S×S) is not homotopy equivalent to a Lie group. Diffeomorphism groups of surfaces and many irreducible 3-manifolds are known to be homotopy equivalent to Lie groups (often discrete groups in fact), and S×S is the simplest manifold for which this is not true. Via the Smale conjecture, proved in [H], the calculation of the homotopy type of Diff(S×S) reduces easily to a problem about making families of 2 spheres in S×S disjoint. To state the problem in slightly more generality, let M be a connected 3 manifold containing a sphere S ⊂ M that does not bound a ball in M , and let S×[−1,1] ⊂ M be a bicollar neighborhood of this S . Let E be the space of smooth embedding f :S2→M3 whose image does not bound a ball, and let E′ be the subspace of embeddings f for which there exists x ∈ [−1,1] with f(S) disjoint from {x}×S . The result we need is: