The Dyer–Lashof algebra and the Steenrod algebra for generalized homology and cohomology

V. A. Smirnov

Sbornik. Mathematics, Tome 190 (1999) no. 12, pp. 1807-1842

Résumé

An analogue $\mathbb R$ of the Dyer–Lashof algebra $R$ and an analogue $\mathbb A$ of the Steenrod algebra $A$ are defined for generalized homology and cohomology theories. It is shown that if there is an $E_\infty$-multiplicative structure on a spectrum $\mathbb H$, then on the corresponding generalized cohomology $\mathbb H^*(X)$ of a topological space $X$ there is an action $\mathbb A\otimes \mathbb H^*(X)\to \mathbb H^*(X)$ of the Steenrod algebra, while if the space $X$ is an $E_\infty$-space, then on the generalized homology $\mathbb H^*(X)$ there is an action $\mathbb R\otimes \mathbb H_*(X)\to \mathbb H_*(X)$ of the Dyer–Lashof algebra. These actions are computed for cobordism of topological spaces. A connection is established between the Steenrod operations and the Landweber–Novikov operations.

Bibliographie

[1] Dyer E., Lashof R. K., “Homology of iterated loop spaces”, Amer. J. Math., 84:1 (1962), 35–88 | DOI | MR | Zbl

[2] Stinrod N., Epstein D., Kogomologicheskie operatsii, Nauka, M., 1983 | MR

[3] Stong R. E., Notes on cobordism theory, Princeton Univ. Press, Princeton, 1968 | MR | Zbl

[4] Landweber P. S., “Cobordism operations and Hopf algebras”, Trans. Amer. Math. Soc., 129 (1967), 94–110 | DOI | MR | Zbl

[5] Novikov S. P., “Koltsa operatsii i spektralnye posledovatelnosti tipa Adamsa v ekstraordinarnykh teoriyakh kogomologii. $U$-kobordizmy i $k$-teoriya”, Dokl. AN SSSR, 172:1 (1967), 33–36 | MR | Zbl

[6] May J. P., The geometry of iterated loop spaces, Lecture Notes in Math., 271, Springer-Verlag, Berlin, 1972 | MR | Zbl

[7] Smirnov V. A., “O kotsepnom komplekse topologicheskogo prostranstva”, Matem. sb., 115 (157) (1981), 146–158 | MR | Zbl

[8] May J. P., Quinn F., Ray N., Tornehave J., $E_\infty$-ring space and $E_\infty$-ring spectra, Lecture Notes in Math., 577, Springer-Verlag, Berlin, 1977 | MR | Zbl

[9] Bruner R. R., May J. P., McClure J. E., Steinberger M., $H_\infty$ ring spectra and their applications, Lecture Notes in Math., 1176, Springer-Verlag, Berlin, 1986 | MR | Zbl

[10] Bisson T., Joyal A., “The Dyer–Lashof algebra in bordism”, C. R. Math. Rep. Acad. Sci. Canada, 17:4 (1995), 135–140 | MR | Zbl

[11] Bisson T., Joyal A., “Nishida relations in bordism and homology”, C. R. Math. Rep. Acad. Sci. Canada, 17:4 (1995), 141–146 | MR | Zbl

[12] Ravenel D. C., Wilson W. S., “The Hopf ring for complex cobordism”, J. Pure Appl. Algebra, 9 (1977), 241–280 | DOI | MR | Zbl

[13] Smirnov V. A., “Vtorichnye operatsii v gomologiyakh operady $E$”, Izv. AN SSSR. Ser. matem., 56:2 (1992), 449–468

[14] Milgram R. J., “Iterated loop spaces”, Ann. of Math. (2), 84:3 (1966), 386–403 | DOI | MR | Zbl

[15] Priddy S. B., “Dyer–Lashof operations for the classifying spaces of certain matrix groups”, Quart. J. Math. Oxford Ser. (2), 26 (1975), 179–193 | DOI | MR | Zbl

[16] Kochman S. O., “Homology of the classical groups over the Dyer–Lashof algebra”, Trans. Amer. Math. Soc., 185 (1973), 83–136 | DOI | MR

[17] Dieck T., “Steenrod-Operationen in Kobordismen-Theorien”, Math. Z., 107 (1968), 380–401 | DOI | MR | Zbl