Periodic phenomena in the classical Adams spectral sequence
HTML articles powered by AMS MathViewer
- by Mark Mahowald and Paul Shick
- Trans. Amer. Math. Soc. 300 (1987), 191-206
- DOI: https://doi.org/10.1090/S0002-9947-1987-0871672-0
- PDF | Request permission
Abstract:
We investigate certain periodic phenomena in the classical Adams sepctral sequence which are related to the polynomial generators ${\nu _n}$ in ${\pi _{\ast }}(\operatorname {BP} )$. We define the notion of a class $a$ in ${\operatorname {Ext} _A}({\mathbf {Z}}/2,{\mathbf {Z}}/2)$ being ${\nu _n}$-periodic or ${\nu _n}$-torsion and prove that classes that are ${\nu _n}$-torsion are also ${\nu _k}$-torsion for all $k$ such that $0 \leqslant k \leqslant n$. This allows us to define a chromatic filtration of ${\operatorname {Ext} _A}({\mathbf {Z}}/2,{\mathbf {Z}}/2)$ paralleling the chromatic filtration of the Novikov spectral sequence ${E_2}$-term given in [13].References
- J. F. Adams, A periodicity theorem in homological algebra, Proc. Cambridge Philos. Soc. 62 (1966), 365โ377. MR 194486, DOI 10.1017/s0305004100039955
- J. F. Adams, On the groups $J(X)$. IV, Topology 5 (1966), 21โ71. MR 198470, DOI 10.1016/0040-9383(66)90004-8
- Nils Andreas Baas, Bordism theories with singularities, Proceedings of the Advanced Study Institute on Algebraic Topology (Aarhus Univ., Aarhus, 1970) Various Publ. Ser., No. 13, Mat. Inst., Aarhus Univ., Aarhus, 1970, pp.ย 1โ16. MR 0346823 M. G. Barratt, Mimeographed notes, Seattle conferences, 1963.
- Edgar H. Brown Jr. and Franklin P. Peterson, A spectrum whose $Z_{p}$ cohomology is the algebra of reduced $p^{th}$ powers, Topology 5 (1966), 149โ154. MR 192494, DOI 10.1016/0040-9383(66)90015-2
- Donald M. Davis and Mark Mahowald, $v_{1}$- and $v_{2}$-periodicity in stable homotopy theory, Amer. J. Math. 103 (1981), no.ย 4, 615โ659. MR 623131, DOI 10.2307/2374044 โ, Ext over the subalgebra ${A_2}$ of the Steenrod algebra for stunted projected spaces, Canad. Math. Soc. 2 (1982), 297-342.
- P. J. Hilton and U. Stammbach, A course in homological algebra, 2nd ed., Graduate Texts in Mathematics, vol. 4, Springer-Verlag, New York, 1997. MR 1438546, DOI 10.1007/978-1-4419-8566-8
- David Copeland Johnson and Zen-ichi Yosimura, Torsion in Brown-Peterson homology and Hurewicz homomorphisms, Osaka Math. J. 17 (1980), no.ย 1, 117โ136. MR 558323
- Mark Mahowald, The primary $v_{2}$-periodic family, Math. Z. 177 (1981), no.ย 3, 381โ393. MR 618203, DOI 10.1007/BF01162070
- W. Lellmann and M. Mahowald, A generalization of the lambda algebra, Math. Z. 192 (1986), no.ย 2, 243โ251. MR 840827, DOI 10.1007/BF01179426
- Wen Hsiung Lin, Cohomology of sub-Hopf-algebras of the Steenrod algebra, J. Pure Appl. Algebra 10 (1977/78), no.ย 2, 101โ113. MR 454975, DOI 10.1016/0022-4049(77)90013-5
- 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.ย 3, 469โ516. MR 458423, DOI 10.2307/1971064
- F. P. Peterson, Lectures on cobordism theory, Lectures in Mathematics; Department of Mathematics, Kyoto University, vol. 1, Kinokuniya Book Store Co., Ltd., Tokyo, 1968. Notes by M. Mimura. MR 0234475 P. L. Shick, Thesis, Northwestern University, 1984.
- 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
- Martin C. Tangora, On the cohomology of the Steenrod algebra, Math. Z. 116 (1970), 18โ64. MR 266205, DOI 10.1007/BF01110185
- Clarence Wilkerson, The cohomology algebras of finite-dimensional Hopf algebras, Trans. Amer. Math. Soc. 264 (1981), no.ย 1, 137โ150. MR 597872, DOI 10.1090/S0002-9947-1981-0597872-X
- Arunas Liulevicius, The cohomology of Massey-Peterson algebras, Math. Z. 105 (1968), 226โ256. MR 233358, DOI 10.1007/BF01109902
Bibliographic Information
- © Copyright 1987 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 300 (1987), 191-206
- MSC: Primary 55T15; Secondary 55N22, 55Q45, 55S10
- DOI: https://doi.org/10.1090/S0002-9947-1987-0871672-0
- MathSciNet review: 871672