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Transactions of the American Mathematical Society

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On the structure of $P(n)_\ast P((n))$ for $p=2$

Author: Christian Nassau
Journal: Trans. Amer. Math. Soc. 354 (2002), 1749-1757
MSC (1991): Primary 55N22; Secondary 55P43
Published electronically: January 7, 2002
MathSciNet review: 1881014
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Abstract: We show that $P(n)_\ast(P(n))$ for $p=2$ with its geometrically induced structure maps is not an Hopf algebroid because neither the augmentation $\epsilon$ nor the coproduct $\Delta$are multiplicative. As a consequence the algebra structure of $P(n)_\ast(P(n))$ is slightly different from what was supposed to be the case. We give formulas for $\epsilon(xy)$ and $\Delta(xy)$ and show that the inversion of the formal group of $P(n)$is induced by an antimultiplicative involution $\Xi:P(n)\rightarrow P(n)$. Some consequences for multiplicative and antimultiplicative automorphisms of $K(n)$ for $p=2$ are also discussed.

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  • [KW] R. Kultze and U. Würgler, A note on the algebra $P(n)_\ast(P(n))$ for the prime $2$, Manuscripta Math. 57 (1987), 195-203 MR 88e:55018
  • [M] O. K. Mironov, Multiplications in cobordism theories with singularities and Steenrod - tom Dieck operations, Math. USSR-IZV 13, (1979), 89-106 MR 80d:55005
  • [N] C. Nassau, Eine nichtgeometrische Konstruktion der Spektren $P(n)$, Multiplikative und antimultiplikative Automorphismen von $K(n)$, Diplomarbeit, Johann Wolfgang Goethe-Universität Frankfurt, October 1995
  • [R1] D. Ravenel, Complex cobordism and stable homotopy groups of spheres, Academic Press, Inc., Orlando, 1986 MR 87j:55003
  • [R2] D. Ravenel, Nilpotence and periodicity in stable homotopy theory, Annals of Mathematics Studies 128, Princeton University Press, 1992 MR 94b:55015
  • [W1] U. Würgler, Commutative ring spectra of characteristic $2$, Comment. Math. Helvetici 61 (1986), 33-45 MR 87i:55008
  • [W2] U. Würgler, Morava $K$-theories: A survey in Algebraic Topology, Poznan 1989 Proceedings, Springer Lecture Notes 1474 (1991), 111-138 MR 92j:55007
  • [Y] N. Yagita, On the Steenrod algebra of Morava $K$-theory, J. London Math. Soc. (2), 22 (1980), 423-438 MR 82f:55027

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Additional Information

Christian Nassau
Affiliation: Johann Wolfgang Goethe-Universität Frankfurt, Fachbereich Mathematik, Robert Mayer Strasse 6-8, 60054 Frankfurt, Germany

Keywords: Hopf algebroids, Morava $K$-theory, bordism theory, noncommutative ring spectra
Received by editor(s): July 3, 2000
Published electronically: January 7, 2002
Article copyright: © Copyright 2002 American Mathematical Society

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