Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-24T19:04:17.275Z Has data issue: false hasContentIssue false
  • English
  • Français

Leibniz Rule on Higher Pages of Unstable Spectral Sequences

Part of: Spectral sequences Applied homological algebra and category theory Homotopy groups

Published online by Cambridge University Press:  01 February 2018

Sergei O. Ivanov ,
Roman Mikhailov  and
Jie Wu
Sergei O. Ivanov
Affiliation:
Chebyshev Laboratory, St. Petersburg State University, 14th Line, 29b, St. Petersburg, 199178Russia (ivanov.s.o.1986@gmail.com)
Roman Mikhailov
Affiliation:
Chebyshev Laboratory, St. Petersburg State University, 14th Line, 29b, St. Petersburg, 199178Russia (ivanov.s.o.1986@gmail.com) St. Petersburg Department of Steklov Mathematical Institute, Fontanka 27, St. Petersburg, 191023Russia (rmikhailov@mail.ru)
Jie Wu*
Affiliation:
Department of Mathematics, National University of Singapore, 10 Lower Kent Ridge Road, Singapore119076 (matwuj@nus.edu.sg)
*
*Corresponding author.
Get access Rights & Permissions [Opens in a new window]

Abstract

A natural composition ⊙ on all pages of the lower central series spectral sequence for spheres is defined. Moreover, it is defined for the p-lower central series spectral sequence of a simplicial group. It is proved that the rth differential satisfies a ‘Leibniz rule with suspension’: dr(a ⊙ σ b) = ±dra ⊙ b + a ⊙ dr σ b, where σ is the suspension homomorphism.

Keywords

spectral sequenceLeibniz rulelower central serieshomotopy groups

MSC classification

Primary: 55T15: Adams spectral sequences
Secondary: 55Q40: Homotopy groups of spheres 55Q05: Homotopy groups, general; sets of homotopy classes 55U10: Simplicial sets and complexes
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 61 , Issue 1 , February 2018 , pp. 265 - 282
Copyright
Copyright © Edinburgh Mathematical Society 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1 Bousfield, A. K. and Curtis, E. B., A spectral sequence for the homotopy of nice spaces, Trans. Amer. Math. Soc. 151 (1970), 457–479.Google Scholar
2 Bousfield, A. K., Curtis, E. B., Kan, D. M., Quillen, D. G., Rector, D. L. and Schlesinger, J. W., The mod-p lower central series and the Adams spectral sequence, Topology 5 (1966), 331–342.Google Scholar
3 Rector, D. L., An unstable Adams spectral sequence, Topology 5 (1966), 343–346.Google Scholar
4 Bousfield, A. K. and Kan, D. M., The homotopy spectral sequence of a space with coefficients in a ring, Topology 11 (1972), 79–106.Google Scholar
5 Bousfield, A. K. and Kan, D., Pairings and products in the homotopy spectral sequence, Trans. Amer. Math. Soc. 177 (1973), 319–343.Google Scholar
6 Bousfield, A. K. and Kan, D. M., Homotopy limits, completions and localizations, Lecture Notes in Mathematics, Volume 304 (Springer, Berlin–New York, 1972).Google Scholar
7 Dold, A. and Puppe, D., Homologie nicht-additiver Funktoren. Anwendungen, Ann. Inst. Fourier (Grenoble) 11 (1961), 201–312.Google Scholar
8 Kan, D. M. and Whitehead, G. W., The reduced join of two spectra, Topology 3 (1965), 239–261.CrossRefGoogle Scholar
9 Leibowitz, D., The E1-term of the lower central series spectral sequence for the homotopy of spaces, PhD thesis, Brandeis University (1972).Google Scholar
10 Mac Lane, S., Categories for the working mathematician, 2nd edition, Graduate Texts in Mathematics, Volume 5 (Springer, 1998).Google Scholar
11 Tangora, M., Computing the homology of the lambda algebra, Memoirs of the American Mathematical Society, Volume 337 (American Mathematical Society, 1985).Google Scholar