Operations and Spectral Sequences. I

Author:
James M. Turner

Journal:
Trans. Amer. Math. Soc. **350** (1998), 3815-3835

MSC (1991):
Primary 18G40, 55S05, 55U15; Secondary 18G30, 55S10, 55S12, 55T10, 55T20

DOI:
https://doi.org/10.1090/S0002-9947-98-02254-5

MathSciNet review:
1487634

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Abstract: Using methods developed by W. Singer and J. P. May, we describe a systematic approach to showing that many spectral sequences, determined by a filtration on a complex whose homology has an action of operations, possess a compatible action of the same operations. As a consequence, we obtain W. Singer's result for Steenrod operations on Serre spectral sequence and extend A. Bahri's action of Dyer-Lashof operations on the second quadrant Eilenberg-Moore spectral sequence.

**1.**A. Bahri, Operations in the second quadrant Eilenberg-Moore spectral sequence*J. Pure and Appl. Alg.*27(1983), 207-222 MR**85b:55031****2.**A. K. Bousfield, On the homology spectral sequence of a cosimplicial space,*Amer. J. of Math.*109(1987), 361-394. MR**88j:55017****3.**A. K. Bousfield and D. M. Kan, A second quadrant homotopy spectral sequence,*Trans. A.M.S.*177(1973), 305-318. MR**51:9063****4.**______ ,*Homotopy Limits, Completions, and Localizations*, Lecture Notes in Mathematics 304, Springer-Verlag, 1972. MR**51:1825****5.**F. Cohen, T. Lada, and J. May,*The Homology of Iterated Loop Spaces*, Lecture Notes in Mathematics 533, Springer-Verlag. MR**55:9096****6.**A. Dold, Über die Steenrodschen Kohomologieoperationen,*Ann. of Math.*73(1961), 258-294. MR**23:A646****7.**W. Dwyer, Higher divided squares in second quadrant spectral sequences,*Trans. A.M.S.*260(1980), 437-447. MR**81f:55022****8.**J. P. May, A general algebraic approach to Steenrod operations, The Steenrod Algebra and its Applications, Lecture Notes in Mathematics 168, Springer-Verlag (1970), 153-231. MR**43:6915****9.**______ ,*The Geometry of Iterated Loop Spaces,*Lecture Notes in Mathematics 271, Springer-Verlag, 1972. MR**54:8623b****10.**J. McClure, Private communication, November 1993.**11.**D. Rector, Steenrod operations in the Eilenberg-Moore spectral sequence, Comment. Math. Helv. 45 (1970), 540-552. MR**43:4040****12.**W. Singer, Steenrod squares in spectral sequences I, II,*Trans. A.M.S.*175(1973), 327-336, 337-353. MR**47:7739****13.**L. Smith, On the Kunneth theorem I, Math. Zeit. 116 (1970), 94-140. MR**44:3315****14.**______ , Steenrod squares in spectral sequences: the cohomology of Hopf algebra extensions and of classifying spaces, preprint, Fordham University (1997)**15.**J. Turner, Looping Bousfield-Kan Towers, in preparation**16.**______ , Operations and spectral sequences II, III, in preparation**17.**C. Weibel,*An Introduction to Homological Algebra*, Cambridge Studies in Advanced Mathematics 38, Cambridge University Press, 1995. MR**95f:18001**

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Additional Information

**James M. Turner**

Affiliation:
Department of Mathematics, College of The Holy Cross, One College Street, Worcester, Massachusetts 01610-2395

Address at time of publication:
Department of Mathematics, Calvin College, 3201 Burton Street, S.E., Grand Rapids, Michigan 49546-4388

Email:
jmt@ziplink.net

DOI:
https://doi.org/10.1090/S0002-9947-98-02254-5

Keywords:
Spectral sequences,
Dold algebras,
Steenrod operations,
Dyer-Lashof operations,
cosimplicial spaces,
infinite loop spaces

Received by editor(s):
October 21, 1996

Article copyright:
© Copyright 1998
American Mathematical Society