The Regular Complex in the 𝐵𝑃⟩1⟨-Adams Spectral Sequence

González, Jesús

doi:10.1090/S0002-9947-98-02263-6

The Regular Complex in the $BP\rangle 1 \langle$-Adams Spectral Sequence
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Trans. Amer. Math. Soc. 350 (1998), 2629-2664 Request permission

Abstract:

We give a complete description of the quotient complex $\mathcal {C}$ obtained by dividing out the $\mathbb {F}_{p}$ Eilenberg-Mac Lane wedge summands in the first term of the $BP\langle 1\rangle$–Adams spectral sequence for the sphere spectrum $S^0$. We also give a detailed computation of the cohomology groups $H^{s,t}(\mathcal {C})$ and obtain as a consequence a vanishing line of slope $(p^{2}-p-1)^{-1}$ in their usual $(t-s, s)$ representation. These calculations are interpreted as giving general simple conditions to lift homotopy classes through a $BP\langle 1 \rangle$ resolution of $S^0$.

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, On the groups $J(X)$. II, Topology 3 (1965), 137–171. MR 198468, DOI 10.1016/0040-9383(65)90040-6
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, The Kahn-Priddy theorem, Proc. Cambridge Philos. Soc. 73 (1973), 45–55. MR 310878, DOI 10.1017/s0305004100047459
Saunders MacLane, Steinitz field towers for modular fields, Trans. Amer. Math. Soc. 46 (1939), 23–45. MR 17, DOI 10.1090/S0002-9947-1939-0000017-3
Gunnar Carlsson, On the stable splitting of $b\textrm {o}\wedge b\textrm {o}$ and torsion operations in connective $K$-theory, Pacific J. Math. 87 (1980), no. 2, 283–297. MR 592736
Allan Clark, Homotopy commutativity and the Moore spectral sequence, Pacific J. Math. 15 (1965), 65–74. MR 177416
Donald M. Davis, Odd primary $b\textrm {o}$-resolutions and $K$-theory localization, Illinois J. Math. 30 (1986), no. 1, 79–100. MR 822385
Donald M. Davis, Sam Gitler, and Mark Mahowald, The stable geometric dimension of vector bundles over real projective spaces, Trans. Amer. Math. Soc. 268 (1981), no. 1, 39–61. MR 628445, DOI 10.1090/S0002-9947-1981-0628445-8
Donald M. Davis and Mark Mahowald, Classification of the stable homotopy types of stunted real projective spaces, Pacific J. Math. 125 (1986), no. 2, 335–345. MR 863530
Donald M. Davis and Mark Mahowald, The image of the stable $J$-homomorphism, Topology 28 (1989), no. 1, 39–58. MR 991098, DOI 10.1016/0040-9383(89)90031-1
Samuel Eilenberg and John C. Moore, Limits and spectral sequences, Topology 1 (1962), 1–23. MR 148723, DOI 10.1016/0040-9383(62)90093-9
Jay Goldman and Gian-Carlo Rota, On the foundations of combinatorial theory. IV. Finite vector spaces and Eulerian generating functions, Studies in Appl. Math. 49 (1970), 239–258. MR 265181, DOI 10.1002/sapm1970493239
J. González. Odd primary ${B}{P}\langle 1 \rangle$-resolutions and classification of the stable summands of stunted lens spaces. Trans. Amer. Math. Soc. (to appear).
J. González. A vanishing line in the $BP\langle 1\rangle$-Adams spectral sequence. Submitted to Topology.
David Copeland Johnson and W. Stephen Wilson, Projective dimension and Brown-Peterson homology, Topology 12 (1973), 327–353. MR 334257, DOI 10.1016/0040-9383(73)90027-X
Richard M. Kane, Operations in connective $K$-theory, Mem. Amer. Math. Soc. 34 (1981), no. 254, vi+102. MR 634210, DOI 10.1090/memo/0254
Wolfgang Lellmann, Operations and co-operations in odd-primary connective $K$-theory, J. London Math. Soc. (2) 29 (1984), no. 3, 562–576. MR 754942, DOI 10.1112/jlms/s2-29.3.562
Wolfgang Lellmann and Mark Mahowald, The $b\textrm {o}$-Adams spectral sequence, Trans. Amer. Math. Soc. 300 (1987), no. 2, 593–623. MR 876468, DOI 10.1090/S0002-9947-1987-0876468-1
Mark Mahowald, $b\textrm {o}$-resolutions, Pacific J. Math. 92 (1981), no. 2, 365–383. MR 618072
Mark Mahowald, The image of $J$ in the $EHP$ sequence, Ann. of Math. (2) 116 (1982), no. 1, 65–112. MR 662118, DOI 10.2307/2007048
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
H. R. Margolis, Eilenberg-Mac Lane spectra, Proc. Amer. Math. Soc. 43 (1974), 409–415. MR 341488, DOI 10.1090/S0002-9939-1974-0341488-9
H. R. Margolis, Spectra and the Steenrod algebra, North-Holland Mathematical Library, vol. 29, North-Holland Publishing Co., Amsterdam, 1983. Modules over the Steenrod algebra and the stable homotopy category. MR 738973
Donald Davis (ed.), Reunión sobre teoría de homotopía, Notas de Matemática y Simposia [Mathematical Notes and Symposia], vol. 1, Sociedad Matemática Mexicana, México, 1975. MR 761715
Haynes R. Miller, On relations between Adams spectral sequences, with an application to the stable homotopy of a Moore space, J. Pure Appl. Algebra 20 (1981), no. 3, 287–312. MR 604321, DOI 10.1016/0022-4049(81)90064-5
Séminaire Henri Cartan, 12ième année: 1959/60. Périodicité des groupes d’homotopie stables des groupes classiques, d’après Bott, Secrétariat Mathématique, École Normale Supérieure, Paris, 1961 (French). Deux fascicules; Deuxième édition, corrigée. MR 0157863
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
W. J. Wong, Powers of a prime dividing binomial coefficients, Amer. Math. Monthly 96 (1989), no. 6, 513–517. MR 999418, DOI 10.2307/2323974