Sheared algebra maps and operation bialgebras for mod 2 homology and cohomology

Pengelley, David; Williams, Frank

doi:10.1090/S0002-9947-99-02468-X

Sheared algebra maps and operation bialgebras for mod 2 homology and cohomology
HTML articles powered by AMS MathViewer

by David J. Pengelley and Frank Williams PDF

Trans. Amer. Math. Soc. 352 (2000), 1453-1492 Request permission

Abstract:

The mod 2 Steenrod algebra $\mathcal {A}$ and Dyer-Lashof algebra $\mathcal {R}$ have both striking similarities and differences arising from their common origins in “lower-indexed” algebraic operations. These algebraic operations and their relations generate a bigraded bialgebra $\mathcal {K}$, whose module actions are equivalent to, but quite different from, those of $\mathcal {A}$ and $\mathcal {R}$. The exact relationships emerge as “sheared algebra bijections”, which also illuminate the role of the cohomology of $\mathcal {K}$. As a bialgebra, $\mathcal {K}^{*}$ has a particularly attractive and potentially useful structure, providing a bridge between those of $\mathcal {A^{*}}$ and $\mathcal {R^{*}}$, and suggesting possible applications to the Miller spectral sequence and the $\mathcal {A}$ structure of Dickson algebras.

References

John Frank Adams, Infinite loop spaces, Annals of Mathematics Studies, No. 90, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1978. MR 505692
Saunders MacLane and O. F. G. Schilling, Infinite number fields with Noether ideal theories, Amer. J. Math. 61 (1939), 771–782. MR 19, DOI 10.2307/2371335
Saunders MacLane and O. F. G. Schilling, Infinite number fields with Noether ideal theories, Amer. J. Math. 61 (1939), 771–782. MR 19, DOI 10.2307/2371335
Terrence P. Bisson and André Joyal, $Q$-rings and the homology of the symmetric groups, Operads: Proceedings of Renaissance Conferences (Hartford, CT/Luminy, 1995) Contemp. Math., vol. 202, Amer. Math. Soc., Providence, RI, 1997, pp. 235–286. MR 1436923, DOI 10.1090/conm/202/02591
T.P. Bisson, D.J. Pengelley, F. Williams, Stabilizing the lower operations for mod two cohomology, in Homotopy Invariant Algebraic Structures: A Conference in Honor of J. Michael Boardman (ed. J. P. Meyer, J. Morava, and W. J. Wilson), Contemporary Mathematics, to appear.
S. R. Bullett and I. G. Macdonald, On the Adem relations, Topology 21 (1982), no. 3, 329–332. MR 649764, DOI 10.1016/0040-9383(82)90015-5
H. E. A. Campbell, F. P. Peterson, and P. S. Selick, Self-maps of loop spaces. I, Trans. Amer. Math. Soc. 293 (1986), no. 1, 1–39. MR 814910, DOI 10.1090/S0002-9947-1986-0814910-1
H. E. A. Campbell, F. P. Peterson, and P. S. Selick, Self-maps of loop spaces. I, Trans. Amer. Math. Soc. 293 (1986), no. 1, 1–39. MR 814910, DOI 10.1090/S0002-9947-1986-0814910-1
Frederick R. Cohen, Thomas J. Lada, and J. Peter May, The homology of iterated loop spaces, Lecture Notes in Mathematics, Vol. 533, Springer-Verlag, Berlin-New York, 1976. MR 0436146
Eldon Dyer and R. K. Lashof, Homology of iterated loop spaces, Amer. J. Math. 84 (1962), 35–88. MR 141112, DOI 10.2307/2372804
V. Giambalvo, F.P. Peterson, $\mathcal {A}$-generators for ideals in the Dickson algebra, to appear.
V. Giambalvo, David J. Pengelley, and Douglas C. Ravenel, A fractal-like algebraic splitting of the classifying space for vector bundles, Trans. Amer. Math. Soc. 307 (1988), no. 2, 433–455. MR 940211, DOI 10.1090/S0002-9947-1988-0940211-9
Nguyễn H. V. Hu’ng and Franklin P. Peterson, $\scr A$-generators for the Dickson algebra, Trans. Amer. Math. Soc. 347 (1995), no. 12, 4687–4728. MR 1316852, DOI 10.1090/S0002-9947-1995-1316852-X
Thomas J. Hunter, On the homology spectral sequence for topological Hochschild homology, Trans. Amer. Math. Soc. 348 (1996), no. 10, 3941–3953. MR 1376548, DOI 10.1090/S0002-9947-96-01742-4
David Kraines and Thomas Lada, The cohomology of the Dyer-Lashof algebra, Proceedings of the Northwestern Homotopy Theory Conference (Evanston, Ill., 1982) Contemp. Math., vol. 19, Amer. Math. Soc., Providence, RI, 1983, pp. 145–152. MR 711048, DOI 10.1090/conm/019/711048
David Kraines and Thomas Lada, Applications of the Miller spectral sequence, Current trends in algebraic topology, Part 1 (London, Ont., 1981) CMS Conf. Proc., vol. 2, Amer. Math. Soc., Providence, R.I., 1982, pp. 479–497. MR 686131
James P. Lin, Steenrod connections and connectivity in $H$-spaces, Mem. Amer. Math. Soc. 68 (1987), no. 369, vi+87. MR 897473, DOI 10.1090/memo/0369
Luciano Lomonaco, A phenomenon of reciprocity in the universal Steenrod algebra, Trans. Amer. Math. Soc. 330 (1992), no. 2, 813–821. MR 1044963, DOI 10.1090/S0002-9947-1992-1044963-2
Ib Madsen, On the action of the Dyer-Lashof algebra in $H_{\ast }(G)$, Pacific J. Math. 60 (1975), no. 1, 235–275. MR 388392
Ib Madsen and R. James Milgram, The classifying spaces for surgery and cobordism of manifolds, Annals of Mathematics Studies, No. 92, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1979. MR 548575
J. Peter May, A general algebraic approach to Steenrod operations, The Steenrod Algebra and its Applications (Proc. Conf. to Celebrate N. E. Steenrod’s Sixtieth Birthday, Battelle Memorial Inst., Columbus, Ohio, 1970) Lecture Notes in Mathematics, Vol. 168, Springer, Berlin, 1970, pp. 153–231. MR 0281196
J. Peter May, Homology operations on infinite loop spaces, Algebraic topology (Proc. Sympos. Pure Math., Vol. XXII, Univ. Wisconsin, Madison, Wis., 1970) Amer. Math. Soc., Providence, R.I., 1971, pp. 171–185. MR 0319195
Haynes Miller, A spectral sequence for the homology of an infinite delooping, Pacific J. Math. 79 (1978), no. 1, 139–155. MR 526673
John Milnor, The Steenrod algebra and its dual, Ann. of Math. (2) 67 (1958), 150–171. MR 99653, DOI 10.2307/1969932
Huỳnh Mui, Modular invariant theory and cohomology algebras of symmetric groups, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 22 (1975), no. 3, 319–369. MR 422451
Goro Nishida, Cohomology operations in iterated loop spaces, Proc. Japan Acad. 44 (1968), 104–109. MR 240811
D.J. Pengelley, F. P. Peterson, F. Williams, A global structure theorem for the mod two Dickson algebras, and unstable cyclic modules over the Steenrod and Kudo-Araki-May algebras, to appear.
D.J. Pengelley, F. Williams, Limits of algebras with shifting and a relationship between the Steenrod and Dyer-Lashof algebras, to appear.
Stewart B. Priddy, Koszul resolutions, Trans. Amer. Math. Soc. 152 (1970), 39–60. MR 265437, DOI 10.1090/S0002-9947-1970-0265437-8
M. M. Postnikov, The lower Steenrod algebra, Uspekhi Mat. Nauk 49 (1994), no. 3(297), 187–188 (Russian); English transl., Russian Math. Surveys 49 (1994), no. 3, 192–193. MR 1289398, DOI 10.1070/RM1994v049n03ABEH002265
V. A. Smirnov, Radical Hopf algebras and the Steenrod algebra, Uspekhi Mat. Nauk 42 (1987), no. 2(254), 245–246 (Russian). MR 898642
P. Hebroni, Sur les inverses des éléments dérivables dans un anneau abstrait, C. R. Acad. Sci. Paris 209 (1939), 285–287 (French). MR 14
Saunders MacLane and O. F. G. Schilling, Infinite number fields with Noether ideal theories, Amer. J. Math. 61 (1939), 771–782. MR 19, DOI 10.2307/2371335
N. E. Steenrod, Cohomology operations, Annals of Mathematics Studies, No. 50, Princeton University Press, Princeton, N.J., 1962. Lectures by N. E. Steenrod written and revised by D. B. A. Epstein. MR 0145525
Richard Steiner, Homology operations and power series, Glasgow Math. J. 24 (1983), no. 2, 161–168. MR 706145, DOI 10.1017/S0017089500005231
Clarence Wilkerson, A primer on the Dickson invariants, Proceedings of the Northwestern Homotopy Theory Conference (Evanston, Ill., 1982) Contemp. Math., vol. 19, Amer. Math. Soc., Providence, RI, 1983, pp. 421–434. MR 711066, DOI 10.1090/conm/019/711066