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Dualization of the Hopf algebra of secondary cohomology operations and the Adams spectral sequence

Published online by Cambridge University Press:  19 January 2011

Hans-Joachim Baues  and
Mamuka Jibladze
Hans-Joachim Baues
Affiliation:
Max-Planck-Institut für Mathematik, Vivatsgasse 7, D-53111 Bonn, Germany, baues@mpim-bonn.mpg.de
Mamuka Jibladze
Affiliation:
Razmadze Mathematical Institute, Alexidze st. 1, Tbilisi 0193, Georgia and Max-Planck-Institut für Mathematik, Vivatsgasse 7, D-53111 Bonn, Germany, jib@rmi.acnet.ge
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Abstract

We describe the dualization of the algebra of secondary cohomology operations in terms of generators extending the Milnor dual of the Steenrod algebra. In this way we obtain explicit formulæ for the computation of the E3-term of the Adams spectral sequence converging to the stable homotopy groups of spheres.

Keywords

Adams spectral sequencesecondary cohomology operation2-categorySteenrod algebradualization of Hopf algebra
Type
Research Article
Information
Journal of K-Theory , Volume 7 , Issue 2 , April 2011 , pp. 203 - 347
Copyright
Copyright © ISOPP 2010

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References

[1]:Adams, J. F., On the nonexistence of elements of Hopf invariant one, Bull. Amer. Math. Soc. 64 (1958), 279–282.CrossRefGoogle Scholar
[2]:Adams, J. F., On the structure and applications of the Steenrod algebra, Comment. Math. Helv. 32 (1958), 180–214.CrossRefGoogle Scholar
[3]:Baues, Hans-Joachim, The algebra of secondary cohomology operations, Progress in Mathematics, vol. 247, Birkhäuser, 2006.Google Scholar
[4]:Baues, Hans-Joachim and Jibladze, Mamuka, Secondary derived functors and the Adams spectral sequence, Topology 45 (2006), no. 2, 295–324.CrossRefGoogle Scholar
[5]:Baues, Hans-Joachim and Jibladze, Mamuka, The computation of the E3-term of the Adams spectral sequence, available at arXiv:math.AT/0407045.Google Scholar
[6]:Baues, Hans-Joachim and Jibladze, Mamuka, Classification of abelian track categories, K-Theory 25 (2002), no. 3, 299–311.CrossRefGoogle Scholar
[7]:Baues, Hans-Joachim, Jibladze, Mamuka, and Pirashvili, Teimuraz, Strengthening track theories, available at arXiv:math.CT/0307185.Google Scholar
[8]:Baues, Hans-Joachim and Wirsching, Günther, Cohomology of small categories, J. Pure Appl. Algebra 38 (1985), no. 2-3, 187–211.CrossRefGoogle Scholar
[9]:Bruner, Robert R., Calculation of large Ext modules, Computers in geometry and topology (Chicago, IL,1986), 1989, pp. 79–104.Google Scholar
[10]:Bruner, Robert R., A new differential in the Adams spectral sequence, Topology 23 (1984), no. 3, 271–276.CrossRefGoogle Scholar
[11]:Hazewinkel, Michiel, The algebra of quasi-symmetric functions is free over the integers, Adv. Math. 164 (2001), no. 2, 283–300.CrossRefGoogle Scholar
[12]:Hovey, Mark, Shipley, Brooke, and Smith, Jeff, Symmetric Spectra, J. AMS 13 (1999), no. 1, 149–208.Google Scholar
[13]:Kristensen, Leif, Massey products in Steenrod's algebra, Proc. Advanced Study Inst. on Algebraic Topology (Aarhus, 1970), Vol. II, 1970, pp. 240–255.Google Scholar
[14]:Meyer, Jean-Pierre, Acyclic models for multicomplexes, Duke Math. J. 45 (1978), no. 1, 67–85.CrossRefGoogle Scholar
[15]:Milnor, John, The Steenrod algebra and its dual, Ann. of Math. (2) 67 (1958), 150–171.CrossRefGoogle Scholar
[16]:Nassau, Christian, Ein neuer Algorithmus zur Untersuchung der Kohomologie der Steenrod-Algebra, Frankfurt am Main, 2001. thesis (available online at http://www.nullhomotopie.de/papers/diss.ps.gz).CrossRefGoogle Scholar
[17]:Ravenel, Douglas C., Complex cobordism and stable homotopy groups of spheres, AMS Chelsea Publishing, American Mathematical Society, University of Rochester - AMS, 2004.Google Scholar
[18]:Sloane, N. J. A., On-Line Encyclopedia of Integer Sequences. http://www.research.att.com/~njas/sequences/A059966Google Scholar