Abstract
Both Adams spectral sequence and Adams–Novikov spectral sequence converge to the stable homotopy groups of sphere π*(S). Suppose an element x in the E2-term of the Adams–Novikov spectral sequence converges to a homotopy element in π*(S). In this paper we determine that the algebraic representative \({\tilde x}\) in the E2-term of the Adams spectral sequence converges to the same homotopy element under the conditions related to the Novikov weight and homological dimension.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Adams J.F., On the structure and applications of the Steenrod algebra. Comment. Math. Helv., 1958, 32: 180–214
Andrews M.J., The v1-periodic part of the Adams spectral sequence at an odd prime. Ph.D. Thesis, Cambridge, MA: Massachusetts Institute of Technology, 2015
Brown E.H., Peterson F.P., A spectrum whose Zp-cohomology is the algebra of reduced pth powers. Topology, 1966, 5: 149–154
Johnson D.C., Miller H.R., Wilson W.S., Zahler R.S., Boundary homomorphisms in the generalized Adams spectral sequence and the nontriviality of infinitely many γt in stable homotop. In: Conference on Homotopy Theory (Evanston, Ill., 1974), Notas Mat. Simpos., 1, México: Soc. Mat. Mex., 1975, 47–63
Miller H.R., Some algebraic aspects of the Adams–Novikov spectral sequence. Ph.D. Thesis, Princeton, NJ: Princeton University, 1975
Novikov S.P., The methods of algebraic topology from the viewpoint of cobordism theories. Math. USSR-Izv., 1967, 1(4): 827–913
Ravenel D.C., Complex Cobordism and Stable Homotopy Groups of Spheres. Pure and Applied Mathematics, 121, Orlando, FL: Academic Press, Inc., 1986
Wang X., Zheng Q., The convergence of \(\tilde \alpha _s^{(n)}{h_0}{h_k}\). Sci. China Ser. A, 1998, 41(6): 622–628
Zahler R.S., The Adams–Novikov spectral sequence for the spheres. Ann. of Math. (2), 1972, 96: 480–504
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of Interest The author declares no conflict of interest.
Rights and permissions
About this article
Cite this article
Shen, W. The Convergences of Part of Algebraic Representatives. Front. Math 19, 171–180 (2024). https://doi.org/10.1007/s11464-022-0194-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11464-022-0194-z