대한수학회지 (Journal of the Korean Mathematical Society)

  • 제43권4호
  • /
  • Pages.783-801
  • /
  • 2006
  • /
  • 0304-9914(pISSN)
  • /
  • 2234-3008(eISSN)
대한수학회 (Korean Mathematical Society)
권호 표지
DOI QR코드

DOI QR Code

NON-TRIVIALITY OF TWO HOMOTOPY ELEMENTS IN π*S

  • Liu Xiugui (School of Mathematical Sciences and LPMC Nankai University)
  • 발행 : 2006.07.01
PDF 다운로드
⟨ 이전 논문 다음 논문 ⟩

초록

Let A be the mod p Steenrod algebra for p an arbitrary odd prime and S the sphere spectrum localized at p. In this paper, some useful propositions about the May spectral sequence are first given, and then, two new nontrivial homotopy elements ${\alpha}_1{\jmath}{\xi}_n\;(p{\geq}5,n\;{\geq}\;3)\;and\;{\gamma}_s{\alpha}_1{\jmath}{\xi}_n\;(p\;{\geq}\;7,\;n\;{\geq}\;4)$ are detected in the stable homotopy groups of spheres, where ${\xi}_n\;{\in}\;{\pi}_{p^nq+pq-2}M$ is obtained in [2]. The new ones are of degree 2(p - 1)($p^n+p+1$) - 4 and 2(p - 1)($p^n+sp^2$ + sp + (s - 1)) - 7 and are represented up to nonzero scalar by $b_0h_0h_n,\;b_0h_0h_n\tilde{\gamma}_s\;{\neq}\;0\;{\in}\;Ext^{*,*}_A^(Z_p,\;Z_p)$ in the Adams spectral sequence respectively, where $3\;{\leq}\;s\;<\;p-2$.

키워드

참고문헌

  1. Ralph L. Cohen, Odd primary families in stable homotopy theory, Mem. Amer. Math. Soc. 30 (1981), no. 242, viii+92pp
  2. J. K. Lin, A new family of filtration three in the stable homotopy of spheres, Hiroshima Math. J. 31 (2001), no. 3, 477-492
  3. J. K. Lin and Q. B. Zheng, A new family of filtration seven in the stable homotopy of spheres, Hiroshima Math. J. 28 (1998), no. 2, 183-205
  4. J. K. Lin and Q. B. Zheng, New families in the stable homotopy of spheres revisited, Acta Math. Sin. (Engl. Ser.) 18 (2002), no. 1, 95-106 https://doi.org/10.1007/s101140100143
  5. X. G. Liu, A nontrivial product in the stable Homotopy groups of spheres, Sci. China Ser. A 47 (2004), no. 6, 831-841 https://doi.org/10.1360/03ys0111
  6. A. Liulevicius, The factorizations of cyclic reduced powers by secondary cohomol- ogy operations, Mem. Amer. Math. Soc. No. 42. 1962, 112 pp
  7. H. R. Miller, D. C. Ravenel, and W. S. Wilson, Periodic phenomena in the Adams-Novikov spectral sequence, Ann. Math.(2) 106 (1977), no. 3, 469-516 https://doi.org/10.2307/1971064
  8. D. C. Ravenel, Complex cobordism and stable Homotopy groups of spheres, Academic Press, Orlando, 1986
  9. X. J. Wang, Ordered cochain complex and $H^4$;*($A_p$) for $p\;\geq\;5$, Beijing Mathematics 1 (1995), 80-99
  10. X. G. Zhou, Higher cohomology operations that detect homotopy classes, Lecture Notes in Mathematics. No. 1370, Springer-verlag, New York-Berlin, 1989

피인용 문헌

  1. A Four-filtrated may spectral sequence and its applications vol.24, pp.9, 2008, https://doi.org/10.1007/s10114-008-6219-z
  2. Detection of some elements in the stable homotopy groups of spheres vol.29, pp.3, 2008, https://doi.org/10.1007/s11401-006-0519-3