Bordism and regular homotopy of low-dimensional immersions
Bordism and regular homotopy of low-dimensional immersions
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低维浸没的 Bordism 和正则同伦
DOI：
10.2140/pjm.1992.156.155
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发表时间：
1992
影响因子：
0.6
通讯作者：
J. Hughes
中科院分区：
文献类型：
--
作者：
J. Hughes
本文研究了低维浸入的几何特征。斯梅尔问，在他的论文浸入的/c-领域的Rn，什么是明确的发电机组的定期同伦类浸入？我们对R4和R5中的3-球面回答这个问题。对于R4中的S3，答案是：定理。S2在R3中的标准外翻（Froissart-Morin）作为一个轨道，在R4中有一个S2 x /的浸入，其末端嵌入S2 s。每一个都在R4中绑定一个3球。用这3个球覆盖轨道产生浸没K：S 3 -· R4。执行两次外翻和封顶给出了一个浸入E：S 3 -> R 4。浸入E和K生成S 3在R 4中浸入的正则同伦类群。
In this paper we study the geometric characteristics of low-dimensional immersions. Smale asked, in his paper on immersions of the /c-sphere in Rn , what are explicit generators for the groups of regular homotopy classes of immersions? We answer this for the 3-sphere in R4 and R5. For S3 in R4 , the answer is: THEOREM. The standard (Froissart-Morin) eversion of S2 in R3 has, as a track, an immersion of S 2 x / in R 4 whose ends are embedded S2s. Each of these bounds a 3-ball in R 4. Capping off the track with these 3-balls yields an immersion K: S 3 —• R4 . Performing the eversion twice and capping off gives an immersion E: S 3 —> R 4 . The immersions E and K generate the group of regular homotopy classes of immersions of S 3 in R 4 .