The nilpotence height of $Sq^{2^n}$
HTML articles powered by AMS MathViewer
- by G. Walker and R. M. W. Wood PDF
- Proc. Amer. Math. Soc. 124 (1996), 1291-1295 Request permission
Abstract:
A 20-year-old conjecture about the mod 2 Steenrod algebra $A$, namely that the element $Sq^{2^n}$ has nilpotence height $2n+2$, is proved. The proof uses formulae of D. M. Davis and J. H. Silverman to obtain commutation relations involving βatomicβ $Sq^i$ and the canonical antiautomorphism of $A$, together with a βstrippingβ technique for obtaining new relations in $A$ from old. This construction goes back to Kristensen [Math. Scand. 16 (1965), 97β115].References
- J. F. Adams, letter to D. M. Davis, February 1985.
- D. Arnon, Monomial bases in the Steenrod algebra, J. Pure Appl. Algebra 96 (1994), 215β223.
- D. P. Carlisle and R. M. W. Wood, On an ideal conjecture in the Steenrod algebra, preprint 1994. (Former title: Facts and fancies about relations in the Steenrod algebra.)
- Donald M. Davis, The antiautomorphism of the Steenrod algebra, Proc. Amer. Math. Soc. 44 (1974), 235β236. MR 328934, DOI 10.1090/S0002-9939-1974-0328934-1
- D. M. Davis, On the height of $Sq^{2^n}$, preprint 1985.
- V. Giambalvo and F. Peterson, On the height of $Sq^{2^n}$, preprint, MIT 1994.
- Leif Kristensen, On a Cartan formula for secondary cohomology operations, Math. Scand. 16 (1965), 97β115. MR 196740, DOI 10.7146/math.scand.a-10751
- K. G. Monks, Nilpotence in the Steenrod algebra, Bol. Soc. Mat. Mexicana 37 (1992), 401β416.
- K. G. Monks, Status report: On the height of $Sq^{2^n}$, Preprint, Univ. of Scranton, Pennsylvania, 1991.
- K. G. Monks, The nilpotence height of $P^s_t$, Proc. Amer. Math. Soc. 124 (1996), 1297β1303.
- John Milnor, The Steenrod algebra and its dual, Ann. of Math. (2) 67 (1958), 150β171. MR 99653, DOI 10.2307/1969932
- Judith H. Silverman, Conjugation and excess in the Steenrod algebra, Proc. Amer. Math. Soc. 119 (1993), no.Β 2, 657β661. MR 1152292, DOI 10.1090/S0002-9939-1993-1152292-8
- Philip D. Straffin Jr., Identities for conjugation in the Steenrod algebra, Proc. Amer. Math. Soc. 49 (1975), 253β255. MR 380796, DOI 10.1090/S0002-9939-1975-0380796-3
- R. M. W. Wood, A note on bases and relations in the Steenrod algebra, preprint 1993, Bull. London Math. Soc. 27 (1995), 380β386.
Additional Information
- G. Walker
- Affiliation: Department of Mathematics, University of Manchester, Manchester M13 9PL, United Kingdom
- Email: grant@ma.man.ac.uk
- R. M. W. Wood
- Affiliation: Department of Mathematics, University of Manchester, Manchester M13 9PL, United Kingdom
- Email: reg@ma.man.ac.uk
- Received by editor(s): June 16, 1992
- Received by editor(s) in revised form: October 7, 1994
- Communicated by: Thomas Goodwillie
- © Copyright 1996 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 124 (1996), 1291-1295
- MSC (1991): Primary 55S10
- DOI: https://doi.org/10.1090/S0002-9939-96-03203-0
- MathSciNet review: 1307571