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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

$ {\bf Z}\sb{(2)}$-knot cobordism in codimension two, and involutions on homotopy spheres


Author: Chao Chu Liang
Journal: Trans. Amer. Math. Soc. 256 (1979), 89-97
MSC: Primary 57R85; Secondary 55M35, 57Q45
MathSciNet review: 546908
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Abstract: Let $ {Z_{(2)}}$ denote the ring of rational 2-adic integers. In this paper, we consider the group $ {\Psi _k}$ of $ {Z_{(2)}}$-cobordism classes of $ {Z_{(2)}} - \operatorname{knot} (\Sigma ^{k + 2}, \,{K^k})$, where $ \Sigma $ is a 1-connected $ {Z_{(2)}}$-sphere $ {Z_{(2)}}$-cobordant to $ {S^{k + 2}}$, and K is a 1-connected $ {Z_{(2)}}$-sphere embedded in $ \Sigma $ with trivial normal bundle. For $ n \geqslant 3$, we will prove that $ {\Psi _{2n}} = 0$ and $ {\Psi _{2n - 1}} = {C_\varepsilon }({Z_{(2)}})$, $ \varepsilon = {( - 1)^n}$. Also, we will show that the group $ \Theta _{4m - 1}^{4m + 1}$ of L-equivalence classes of differentiable involutions on $ (4m + 1)$-homotopy spheres with codimension two fixed point sets defined by Bredon contains infinitely many copies of Z.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1979-0546908-1
PII: S 0002-9947(1979)0546908-1
Keywords: $ {Z_{(2)}}$-knot, $ {Z_{(2)}}$-cobordism, simple $ {Z_{(2)}}$-knot, Seifert manifold, Seifert matrix, normal map, normal cobordism, surgery with coefficient, involutions
Article copyright: © Copyright 1979 American Mathematical Society