Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

The Brown-Peterson homology of Mahowald's $ X\sb k$ spectra


Author: Dung Yung Yan
Journal: Trans. Amer. Math. Soc. 344 (1994), 261-289
MSC: Primary 55Q10
DOI: https://doi.org/10.1090/S0002-9947-1994-1272464-7
MathSciNet review: 1272464
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We compute the Brown-Peterson homology of Mahowald's $ {X_k}$ spectrum which is the Thom spectrum induced from $ \Omega {J_{{2^k} - 1}}{S^2} \to {\Omega ^2}{S^3} - {\text{BO}}$, and the edge homomorphism of the Adams-Novikov spectral sequence for $ {\pi _\ast}({X_k})$. We then compute the nonnilpotent elements of $ {\pi _\ast}({X_k})$.


References [Enhancements On Off] (What's this?)

  • [1] J. F. Adams, Stable homotopy and generalized homology, Univ. of Chicago Press, 1974. MR 0402720 (53:6534)
  • [2] -, On the non-existence of elements of Hopf invariant one, Ann. of Math. (2) 72 (1960), 20-104. MR 0141119 (25:4530)
  • [3] P. F. Baum, On the cohomology of homogeneous space, Topology 7 (1968), 15-38. MR 0219085 (36:2168)
  • [4] J. M. Boardman and R. M. Vogt, Homotopy invariant algebraic structures on topological spaces, Lecture Notes in Math., vol. 347, Springer-Verlag, 1973. MR 0420609 (54:8623a)
  • [5] E. H. Brown and F. P. Peterson, Computation of the unoriented cobordism ring, Proc. Amer. Math. Soc. 55 (1976), 191-192. MR 0394708 (52:15507)
  • [6] R. R. Bruner, J. P. May, J. E. McClure, and M. Steinberger, $ {H_\infty }$ ring spectra and their applications, Lecture Notes in Math., vol. 1176, Springer-Verlag, 1984. MR 836132 (88e:55001)
  • [7] H. Cartan, Démonstration homologique des théorèmes de périodi, Séminaire Cartan, 1959-1960.
  • [8] F. R. Cohen, J. P. May, and L. R. Taylor, $ K(\mathbb{Z},0)$ and $ K(\mathbb{Z}/2,0)$ as Thom spectra, Illinois J. Math. 25 (1981), 99-106. MR 602900 (82h:55008)
  • [9] F. R. Cohen, T. J. Lada, and J. P. May, The homology of iterated loop spaces, Lecture Notes in Math., vol. 533, Springer-Verlag, 1976. MR 0436146 (55:9096)
  • [10] M. C. Crabb and S. A. Mitchell, The loops on $ {\text{U}}(n)/{\text{O}}(n)$ and $ {\text{U}}(2n)/{\text{SP}}(n)$, Math. Proc. Cambridge Philos. Soc. 104 (1988), 95-103. MR 938454 (89h:55019)
  • [11] E. S. Devinatz, A nilpotence theorem in stable homotopy theory, Thesis, M.I.T., 1985.
  • [12] E. S. Devinatz, M. J. Hopkins, and J. H. Smith, Nilpotence and stable homotopy theory. I, Ann. of Math. (2) 128 (1988), 207-241. MR 960945 (89m:55009)
  • [13] W. G. Dwyer, Strong convergence of Eilenberg-Moore spectral sequence, Topology 13 (1974), 255-265. MR 0394663 (52:15464)
  • [14] M. J. Hopkins, Stable decomposition of certain loop spaces, Thesis, Northwestern Univ., 1984.
  • [15] S. O. Kochman, Homology of the classical groups over the Dyer-Lashof algebra, Trans. Amer. Math. Soc. 185 (1973), 83-136. MR 0331386 (48:9719)
  • [16] A. Liulevicius, The cohomology of Massey-Peterson algebras, Math. Z. 105 (1968), 226-256. MR 0233358 (38:1680)
  • [17] I. Madsen and R. J. Milgram, On spherical fiber bundles and their PL reductions, London Math. Soc. Lecture Notes, vol. 11, London Math. Soc., 1974, pp. 43-60. MR 0343286 (49:8028)
  • [18] M. Mahowald, A new infinite family in $ _2\pi _\ast ^S$, Topology 16 (1977), 249-256. MR 0445498 (56:3838)
  • [19] -, Ring spectra which are Thom complexes, Duke Math. J. 46 (1979), 549-559. MR 544245 (81f:55010)
  • [20] H. R. Miller, D. C. Ravenel, and W. S. Wilson, Periodic phenomena in the Adams-Novikov spectral sequence, Ann. of Math. (2) 106 (1977), 469-516. MR 0458423 (56:16626)
  • [21] J. W. Milnor, The Steenrod algebra and its dual, Ann. of Math. (2) 69 (1985).
  • [22] J. W. Milnor and J. C. Moore, On the structure of Hopf algebras, Ann. of Math. (2) 67 (1958), 150-171. MR 0174052 (30:4259)
  • [23] J. W. Milnor and J. D. Stasheff, Characteristic classes, Ann. of Math. Stud., no. 76, Princeton Univ. Press, 1974. MR 0440554 (55:13428)
  • [24] S. A. Mitchell, Power series methods in unoriented cobordism, Contemp. Math., vol. 19, Amer. Math. Soc., Providence, RI, 1983, pp. 247-253. MR 711056 (85e:57040)
  • [25] S. Priddy, $ K(\mathbb{Z}/2)$ as a Thom spectrum, Proc. Amer. Math. Soc. 70 (1978), 207-208. MR 0474271 (57:13918)
  • [26] D. C. Ravenel, Complex cobordism and stable homotopy groups of spheres, Academic Press, 1986. MR 860042 (87j:55003)
  • [27] -, A novice's guide to the Adams-Novikov spectral sequence, Geometric Applications in Homotopy Theory. II, Lecture Notes in Math., vol. 658, Springer-Verlag, pp. 404-475.
  • [28] -, Localization with respect to certain periodic homology theories, Amer. J. Math. 106 (1984), 351-414. MR 737778 (85k:55009)
  • [29] -, The structure of $ {\text{BP}}_ \ast {\text{BP}}$ modulo an invariant prime ideal, Topology 15 (1976), 149-153. MR 0420598 (54:8612)
  • [30] N. E. Steenrod and D. B. A. Epstein, Cohomology operations, Ann. of Math. Stud., no. 50, Princeton Univ. Press, 1962. MR 0145525 (26:3056)
  • [31] W. S. Wilson, Brown-Peterson homology: an introduction and sampler, CBMS Regional Conf. Ser. in Math., no. 48, Amer. Math. Soc., Providence, RI, 1982. MR 655040 (83j:55005)
  • [32] Dung Yung Yan, On the Thom spectra over $ \Omega ({\text{SU}}(n)/{\text{SO}}(n))$, and Mahowald's $ {X_k}$ spectra, Proc. Amer. Math. Soc. 116 (1992), 567-573. MR 1123672 (92m:55007)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 55Q10

Retrieve articles in all journals with MSC: 55Q10


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1994-1272464-7
Article copyright: © Copyright 1994 American Mathematical Society

American Mathematical Society