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| Detection of Some Elements in the Stable Homotopy Groups of Spheres |
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Citation： |
Xiugui LIU.Detection of Some Elements in the Stable Homotopy Groups of Spheres[J].Chinese Annals of Mathematics B,2008,29(3):291~316 |
| Page view： 1266
Net amount： 831 |
Authors： |
Xiugui LIU; |
Foundation： |
Project supported by the National Natural Science Foundation of China (Nos. 10501045, 10771105) and the Fund of the Personnel Division of Nankai University (No. J02017). |
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| Abstract： |
Let A be the mod p Steenrod algebra and S be the sphere spectrum localized at an odd prime p. To determine the stable homotopy groups of spheres \pi_{\ast}S is one of the central problems in homotopy theory. This paper constructs a new nontrivial family of homotopy elements in the stable homotopy groups of spheres \pi_{p^nq+2pq+q-3}S which is of order p and is represented by k_0h_{n} \in \Ext_A^{3,p^nq+2pq+q}(\mathbb{Z}_p,\mathbb{Z}_p) in the Adams spectral sequence, where p\geq 5 is an odd prime, n\geq 3 and q=2(p-1). In the course of the proof, a new family of homotopy elements in \pi_{p^nq+(p+1)q-1}V(1) which is represented by \beta_{\ast}{i^{\prime}}_{\ast}i_{\ast}({h}_n)\in \Ext_A^{2,p^nq+(p+1)q+1}(H^{\ast}V(1),\mathbb{Z}_p) in the Adams sequence is detected. |
Keywords： |
Stable homotopy groups of spheres, Adams spectral sequence, May spectral sequence, Steenrod algebra |
Classification： |
55Q45 |
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