Steenrod and Dyer-Lashof operations on $B\textbf {U}$
HTML articles powered by AMS MathViewer
- by Timothy Lance PDF
- Trans. Amer. Math. Soc. 276 (1983), 497-510 Request permission
Abstract:
This paper describes a simple, fast algorithm for the computation of Steenrod and Dyer-Lashof operations on $BU$. The calculations are carried out in $H^{\ast }(BU,{{\mathbf {Z}}_{(p)}})$ and ${H_{\ast }}(BU,{{\mathbf {Z}}_{(p)}})$ where $p$ local lifts are determined by the values on primitives and Cartan formulas. This algorithm also provides a description of Steenrod and Dyer-Lashof operations on the fiber of any $H$ map (or infinite loop map) $BU \to BU$, and applications to the classifying spaces of surgery which arise in this fashion will appear shortly.References
- George E. Andrews, The theory of partitions, Encyclopedia of Mathematics and its Applications, Vol. 2, Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1976. MR 0557013
- David Mumford, Lectures on curves on an algebraic surface, Annals of Mathematics Studies, No. 59, Princeton University Press, Princeton, N.J., 1966. With a section by G. M. Bergman. MR 0209285, DOI 10.1515/9781400882069
- A. Borel and J.-P. Serre, Groupes de Lie et puissances rΓ©duites de Steenrod, Amer. J. Math. 75 (1953), 409β448 (French). MR 58213, DOI 10.2307/2372495
- Edgar H. Brown Jr. and Franklin P. Peterson, Some remarks about symmetric functions, Proc. Amer. Math. Soc. 60 (1976), 349β352 (1977). MR 433465, DOI 10.1090/S0002-9939-1976-0433465-6
- Edgar H. Brown Jr., Donald M. Davis, and Franklin P. Peterson, The homology of BO and some results about the Steenrod algebra, Math. Proc. Cambridge Philos. Soc. 81 (1977), no.Β 3, 393β398. MR 440550, DOI 10.1017/S0305004100053470
- Harriet Griffin, Elementary theory of numbers, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1954. MR 0064063
- Dale Husemoller, The structure of the Hopf algebra $H_\ast (BU)$ over a Z$_{(p)}$-algebra, Amer. J. Math. 93 (1971), 329β349. MR 286867, DOI 10.2307/2373380
- Stanley O. Kochman, Homology of the classical groups over the Dyer-Lashof algebra, Trans. Amer. Math. Soc. 185 (1973), 83β136. MR 331386, DOI 10.1090/S0002-9947-1973-0331386-2
- Timothy Lance, Local $H$-maps of classifying spaces, Trans. Amer. Math. Soc. 254 (1979), 195β215. MR 539915, DOI 10.1090/S0002-9947-1979-0539915-6 β, The homology of some classifying spaces for surgery (preprint). β, The complex bordism of the Postnikov tower of $N$ (preprint).
- Arunas Liulevicius, On characteristic classes, Aarhus Universitet, Matematisk Institut, Aarhus, 1968. Lectures given at the Nordic Summer School in Mathematics (June 16th-July 6th, 1968). MR 0256409 P. MacMahon, Combinatory analysis, Cambridge Univ. Press, New York, 1915-16.
- John W. Milnor and James D. Stasheff, Characteristic classes, Annals of Mathematics Studies, No. 76, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1974. MR 0440554, DOI 10.1515/9781400881826 D. Moore, On homology operations for the classifying spaces of certain groups, Ph.D. thesis, Northwestern University, 1974.
- S. Mukohda and S. Sawaki, On the $b_p^{k,j}$ coefficient of a certain symmetric function, J. Fac. Sci. Niigata Univ. Ser. I 1 (1954), no.Β 2, 6. MR 91262
- Franklin P. Peterson, A $\textrm {mod}$ $p$ Wu formula, Bol. Soc. Mat. Mexicana (2) 20 (1975), no.Β 2, 56β58. MR 461506
- Stewart Priddy, Dyer-Lashof operations for the classifying spaces of certain matrix groups, Quart. J. Math. Oxford Ser. (2) 26 (1975), no.Β 102, 179β193. MR 375309, DOI 10.1093/qmath/26.1.179
- Earl D. Rainville, Special functions, The Macmillan Company, New York, 1960. MR 0107725
- P. Brian Shay, $\textrm {mod}$ $p$ Wu formulas for the Steenrod algebra and the Dyer-Lashof algebra, Proc. Amer. Math. Soc. 63 (1977), no.Β 2, 339β347. MR 454974, DOI 10.1090/S0002-9939-1977-0454974-0
Additional Information
- © Copyright 1983 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 276 (1983), 497-510
- MSC: Primary 55S10; Secondary 55R45, 55S12
- DOI: https://doi.org/10.1090/S0002-9947-1983-0688957-X
- MathSciNet review: 688957