Embeddings of Z2-homology 3-spheres in R5 up to regular homotopy
Embeddings of Z2-homology 3-spheres in R5 up to regular homotopy
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R5 中 Z2 同源 3-球体的嵌入达到正则同伦
DOI：
10.2140/pjm.2000.193.249
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发表时间：
2000
影响因子：
0.6
通讯作者：
Masamichi Takase
中科院分区：
文献类型：
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作者：
Masamichi Takase
令 Imm[X,Y ] 为流形 X 在流形 Y 中的浸没的正则同伦类的集合，并且 Emb[X,Y ] 表示由包含嵌入的所有正则同伦类组成的 Imm[X,Y ] 的子集。 Smale [6] 给出了 1-1 对应关系（Smale 不变量） s : Imm[Sn,RN ]→ πn(VN,n)，其中 VN,n 是 RN 中所有 n 帧的 Stiefel 流形。 Hirsch [2] 将其推广到任意流形浸入任意流形的情况。这些结果解决了同伦理论中正则同伦类的数量问题，但没有成功地找到每个类的代表或确定哪些类由嵌入表示。根据 Hughes [4]，Imm[Sn,RN ] 在连通和下具有群结构，并且 Smale 不变量实际上给出了群同构。 [4] 给出了 Imm[S3,R4] 和 Imm[S3,R5] 的显式生成器。 Hughes-Melvin [5] 确定 Imm[Sn,Rn+2] 的哪些类由嵌入表示，并证明 Emb[Sn,Rn+2] 在 n == 3 mod 4 时与 Z 同构，否则与 0 同构。此外，[5]证明嵌入 Sn ↪→ Rn+2(n ≡ 3 mod 4) 的正则同伦类可以完全由其定向“Seifert”流形的签名确定。例如，在 n = 3 的情况下，我们有下图：
Let Imm[X,Y ] be the set of regular homotopy classes of immersions of a manifold X in a manifold Y , and Emb[X,Y ] denote the subset of Imm[X,Y ] consisting of all regular homotopy classes containing an embedding. Smale [6] has given a 1-1 correspondence (the Smale invariant) s : Imm[Sn,RN ]→ πn(VN,n), where VN,n is the Stiefel manifold of all n-frames in RN . Hirsch [2] has generalized this to the case of immersions of an arbitrary manifold in an arbitrary manifold. These results solve the problem of the number of regular homotopy classes in terms of homotopy theory, but do not succeed in finding representatives for each class or determining which classes are represented by an embedding. According to Hughes [4], Imm[Sn,RN ] has a group structure under connected sum and the Smale invariant actually gives a group isomorphism. [4] gives explicit generators of Imm[S3,R4] and Imm[S3,R5]. Hughes-Melvin [5] determine which classes of Imm[Sn,Rn+2] are represented by an embedding, and prove that Emb[Sn,Rn+2] is isomorphic to Z if n ≡ 3 mod 4, and to 0 otherwise. Furthermore, [5] proves that the regular homotopy class of an embedding Sn ↪→ Rn+2(n ≡ 3 mod 4) can be completely determined by the signature of its oriented “Seifert” manifold. For example, in the case n = 3, we have the following diagram: