The iterated transfer analogue of the new doomsday conjecture
HTML articles powered by AMS MathViewer
- by Norihiko Minami
- Trans. Amer. Math. Soc. 351 (1999), 2325-2351
- DOI: https://doi.org/10.1090/S0002-9947-99-02037-1
- Published electronically: February 16, 1999
- PDF | Request permission
Abstract:
A strong general restriction is given on the stable Hurewicz image of the classifying spaces of elementary abelian $p$-groups. In particular, this implies the iterated transfer analogue of the new doomsday conjecture.References
- J. F. Adams, On the non-existence of elements of Hopf invariant one, Ann. of Math. (2) 72 (1960), 20–104. MR 141119, DOI 10.2307/1970147
- J. F. Adams, Vector fields on spheres, Ann. of Math. (2) 75 (1962), 603–632. MR 139178, DOI 10.2307/1970213
- J. F. Adams, Lectures on generalised cohomology, Category Theory, Homology Theory and their Applications, III (Battelle Institute Conference, Seattle, Wash., 1968, Vol. Three), Springer, Berlin, 1969, pp. 1–138. MR 0251716
- J. F. Adams and M. F. Atiyah, $K$-theory and the Hopf invariant, Quart. J. Math. Oxford Ser. (2) 17 (1966), 31–38. MR 198460, DOI 10.1093/qmath/17.1.31
- J. F. Adams, J. H. Gunawardena, and H. Miller, The Segal conjecture for elementary abelian $p$-groups, Topology 24 (1985), no. 4, 435–460. MR 816524, DOI 10.1016/0040-9383(85)90014-X
- Donald W. Anderson and Donald M. Davis, A vanishing theorem in homological algebra, Comment. Math. Helv. 48 (1973), 318–327. MR 334207, DOI 10.1007/BF02566125
- Shôrô Araki, Typical formal groups in complex cobordism and $K$-theory, Lectures in Mathematics, Department of Mathematics, Kyoto University, No. 6, Kinokuniya Book Store Co., Ltd., Tokyo, 1973. MR 0375354
- S. Araki, Multiplicative operations in $BP$ cohomology, Osaka J. Math. 12 (1975), 343–356.
- J. Michael Boardman, Modular representations on the homology of powers of real projective space, Algebraic topology (Oaxtepec, 1991) Contemp. Math., vol. 146, Amer. Math. Soc., Providence, RI, 1993, pp. 49–70. MR 1224907, DOI 10.1090/conm/146/01215
- A. K. Bousfield, The localization of spectra with respect to homology, Topology 18 (1979), no. 4, 257–281. MR 551009, DOI 10.1016/0040-9383(79)90018-1
- Gunnar Carlsson, G. B. Segal’s Burnside ring conjecture for $(\textbf {Z}/2)^{k}$, Topology 22 (1983), no. 1, 83–103. MR 682060, DOI 10.1016/0040-9383(83)90046-0
- Gunnar Carlsson, Equivariant stable homotopy and Segal’s Burnside ring conjecture, Ann. of Math. (2) 120 (1984), no. 2, 189–224. MR 763905, DOI 10.2307/2006940
- Donald M. Davis, The BP-coaction for projective spaces, Canadian J. Math. 30 (1978), no. 1, 45–53. MR 478151, DOI 10.4153/CJM-1978-004-9
- Idar Hansen, Primitive and framed elements in $MU_{^*}\textbf {Z}/p$, Math. Z. 157 (1977), no. 1, 43–52. MR 464265, DOI 10.1007/BF01214679
- Michiel Hazewinkel, Formal groups and applications, Pure and Applied Mathematics, vol. 78, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 506881
- David Copeland Johnson, Peter S. Landweber, and Zen-ichi Yosimura, Injective $\textrm {BP}_\ast \textrm {BP}$-comodules and localizations of Brown-Peterson homology, Illinois J. Math. 25 (1981), no. 4, 599–610. MR 630844
- R. H. J. Germay, Généralisation de l’équation de Hesse, Ann. Soc. Sci. Bruxelles Sér. I 59 (1939), 139–144 (French). MR 86
- David Copeland Johnson, W. Stephen Wilson, and Dung Yung Yan, Brown-Peterson homology of elementary $p$-groups. II, Topology Appl. 59 (1994), no. 2, 117–136. MR 1296028, DOI 10.1016/0166-8641(94)90090-6
- David Copeland Johnson and Zen-ichi Yosimura, Torsion in Brown-Peterson homology and Hurewicz homomorphisms, Osaka Math. J. 17 (1980), no. 1, 117–136. MR 558323
- Daniel S. Kahn and Stewart B. Priddy, Applications of the transfer to stable homotopy theory, Bull. Amer. Math. Soc. 78 (1972), 981–987. MR 309109, DOI 10.1090/S0002-9904-1972-13076-3
- M. Kameko, Products of projective spaces as Steenrod modules, Ph. D. Thesis, Johns Hopkins University, May 1990.
- Peter S. Landweber, Künneth formulas for bordism theories, Trans. Amer. Math. Soc. 121 (1966), 242–256. MR 192503, DOI 10.1090/S0002-9947-1966-0192503-7
- P. S. Landweber, Coherence, flatness and cobordism of classifying spaces, Proc. Aarhus Summer Institute of algebraic topology, 1970, pp. 256–269.
- Jean Lannes, Sur les espaces fonctionnels dont la source est le classifiant d’un $p$-groupe abélien élémentaire, Inst. Hautes Études Sci. Publ. Math. 75 (1992), 135–244 (French). With an appendix by Michel Zisman. MR 1179079, DOI 10.1007/BF02699494
- Arunas Liulevicius, The factorization of cyclic reduced powers by secondary cohomology operations, Mem. Amer. Math. Soc. 42 (1962), 112. MR 182001
- Mark Mahowald, A new infinite family in ${}_{2}\pi _{*}{}^s$, Topology 16 (1977), no. 3, 249–256. MR 445498, DOI 10.1016/0040-9383(77)90005-2
- M. E. Mahowald, Some remarks on the Kervaire invariant problem from the homotopy point of view, Algebraic topology (Proc. Sympos. Pure Math., Vol. XXII, Univ. Wisconsin, Madison, Wis., 1970) Amer. Math. Soc., Providence, R.I., 1971, pp. 165–169. MR 0322884
- M. Mahowald and R. James Milgram, Operations which detect Sq4 in connective $K$-theory and their applications, Quart. J. Math. Oxford Ser. (2) 27 (1976), no. 108, 415–432. MR 433453, DOI 10.1093/qmath/27.4.415
- 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
- Haynes Miller, The Sullivan conjecture on maps from classifying spaces, Ann. of Math. (2) 120 (1984), no. 1, 39–87. MR 750716, DOI 10.2307/2007071
- Haynes R. Miller, Douglas C. Ravenel, and W. Stephen Wilson, Periodic phenomena in the Adams-Novikov spectral sequence, Ann. of Math. (2) 106 (1977), no. 3, 469–516. MR 458423, DOI 10.2307/1971064
- John Milnor, The Steenrod algebra and its dual, Ann. of Math. (2) 67 (1958), 150–171. MR 99653, DOI 10.2307/1969932
- Norihiko Minami, On the double transfer, Algebraic topology (Oaxtepec, 1991) Contemp. Math., vol. 146, Amer. Math. Soc., Providence, RI, 1993, pp. 339–347. MR 1224923, DOI 10.1090/conm/146/01231
- Norihiko Minami, The Kervaire invariant one element and the double transfer, Topology 34 (1995), no. 2, 481–488. MR 1318889, DOI 10.1016/0040-9383(94)00030-O
- N. Minami, On the odd primary Adams 2-line elements, preprint.
- Norihiko Minami, The Adams spectral sequence and the triple transfer, Amer. J. Math. 117 (1995), no. 4, 965–985. MR 1342837, DOI 10.2307/2374955
- Lu Ning, Equivalence relations on manifolds, Stud. Cerc. Mat. 36 (1984), no. 2, 105–112 (Romanian, with English summary). MR 764531
- S. P. Novikov, Methods of algebraic topology from the point of view of cobordism theory, Izv. Akad. Nauk SSSR Ser. Mat. 31 (1967), 855–951 (Russian). MR 0221509
- Franklin P. Peterson, $A$-generators for certain polynomial algebras, Math. Proc. Cambridge Philos. Soc. 105 (1989), no. 2, 311–312. MR 974987, DOI 10.1017/S0305004100067803
- Douglas C. Ravenel, Localization with respect to certain periodic homology theories, Amer. J. Math. 106 (1984), no. 2, 351–414. MR 737778, DOI 10.2307/2374308
- Douglas C. Ravenel, Complex cobordism and stable homotopy groups of spheres, Pure and Applied Mathematics, vol. 121, Academic Press, Inc., Orlando, FL, 1986. MR 860042
- Douglas C. Ravenel, The geometric realization of the chromatic resolution, Algebraic topology and algebraic $K$-theory (Princeton, N.J., 1983) Ann. of Math. Stud., vol. 113, Princeton Univ. Press, Princeton, NJ, 1987, pp. 168–179. MR 921477
- Douglas C. Ravenel and W. Stephen Wilson, The Morava $K$-theories of Eilenberg-Mac Lane spaces and the Conner-Floyd conjecture, Amer. J. Math. 102 (1980), no. 4, 691–748. MR 584466, DOI 10.2307/2374093
- F. W. Roush, Transfer in generalized cohomology theories, Ph. D. thesis, Princeton Univ., 1971.
- William M. Singer, The transfer in homological algebra, Math. Z. 202 (1989), no. 4, 493–523. MR 1022818, DOI 10.1007/BF01221587
- William M. Singer, On the action of Steenrod squares on polynomial algebras, Proc. Amer. Math. Soc. 111 (1991), no. 2, 577–583. MR 1045150, DOI 10.1090/S0002-9939-1991-1045150-9
- Sheng Min Chen and Xin Yao Shen, On the action of Steenrod powers on polynomial algebras, Algebraic topology (San Feliu de Guíxols, 1990) Lecture Notes in Math., vol. 1509, Springer, Berlin, 1992, pp. 326–330. MR 1185982, DOI 10.1007/BFb0087522
- W. Stephen Wilson, Brown-Peterson homology: an introduction and sampler, CBMS Regional Conference Series in Mathematics, vol. 48, Conference Board of the Mathematical Sciences, Washington, D.C., 1982. MR 655040
- R. M. W. Wood, Steenrod squares of polynomials and the Peterson conjecture, Math. Proc. Cambridge Philos. Soc. 105 (1989), no. 2, 307–309. MR 974986, DOI 10.1017/S0305004100067797
Bibliographic Information
- Norihiko Minami
- Affiliation: Department of Mathematics, The University of Alabama, Box 870350, Tuscaloosa, Alabama 35487–0350
- Email: norihiko@gp.as.ua.edu, norihiko@euler.math.ua.edu
- Received by editor(s): November 1, 1996
- Published electronically: February 16, 1999
- Additional Notes: This work was partially supported by a University of Alabama Research Grant.
- © Copyright 1999 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 351 (1999), 2325-2351
- MSC (1991): Primary 55Q10, 55Q45, 55T15; Secondary 55R12, 55R35, 55N22, 57R77
- DOI: https://doi.org/10.1090/S0002-9947-99-02037-1
- MathSciNet review: 1443884
Dedicated: Dedicated to Professor Dan Kan on the occasion of his retirement from teaching calculus