Springer Nature is making Coronavirus research free. View research | View latest news | Sign up for updates

Uniqueness of \(\textit{BP} \langle {n} \rangle \)

  • 93 Accesses

  • 2 Citations


Fix a prime number p and an integer \(n \ge 0\). We prove that if a p-complete spectrum X satisfying a mild finiteness condition has the same mod p cohomology as \(\textit{BP} \langle {n} \rangle \) as a module over the Steenrod algebra, then X is weak homotopy equivalent to the p-completion of \(\textit{BP}\langle {n} \rangle \).

This is a preview of subscription content, log in to check access.


  1. 1.

    Adams, J.F., Priddy, S.B.: Uniqueness of \(B{\rm SO}\). Math. Proc. Camb. Philos. Soc. 80(3), 475–509 (1976)

  2. 2.

    Araki, S.: Typical Formal Groups in Complex Cobordism and \(K\)-Theory. Lectures in Mathematics. Kyoto University, No. 6, Department of Mathematics. Kinokuniya Book-Store Co., Ltd, Tokyo (1973)

  3. 3.

    Baas, N.A., Madsen, I.B.: On the realization of certain modules over the Steenrod algebra. Math. Scand. 31, 220–224 (1972)

  4. 4.

    Boardman, J.M.: Conditionally convergent spectral sequences. In: Homotopy invariant algebraic structures (Baltimore, MD, 1998), volume 239 of Contemp. Math., pp 49–84. Amer. Math. Soc., Providence (1999)

  5. 5.

    Bousfield, A.K.: The localization of spectra with respect to homology. Topology 18(4), 257–281 (1979)

  6. 6.

    Hazewinkel, M.: Constructing formal groups. I. The local one dimensional case. J. Pure Appl. Algebra, 9(2):131–149 (1976/77)

  7. 7.

    Lawson, T., Naumann, N.: Commutativity conditions for truncated Brown–Peterson spectra of height 2. J. Topol. 5(1), 137–168 (2012)

  8. 8.

    Lawson, T., Naumann, N.: Strictly commutative realizations of diagrams over the Steenrod algebra and topological modular forms at the prime 2. Int. Math. Res. Not. (IMRN) 10, 2773–2813 (2014)

  9. 9.

    Milnor, J.: The Steenrod algebra and its dual. Ann. Math. 2(67), 150–171 (1958)

  10. 10.

    Quillen, D.: On the formal group laws of unoriented and complex cobordism theory. Bull. Am. Math. Soc. 75, 1293–1298 (1969)

  11. 11.

    Ravenel, D.C.: Complex Cobordism and Stable Homotopy Groups of Spheres. Pure and Applied Mathematics, vol. 121. Academic Press, Orlando (1986)

  12. 12.

    Wilson,W.S.: The \(\Omega \)-spectrum for Brown–Peterson cohomology. I. Comment. Math. Helv. 48, 45–55; corrigendum, ibid. 48 (1973), 194, (1973)

  13. 13.

    Wilson, W.S.: The \(\Omega \)-spectrum for Brown–Peterson cohomology. II. Am. J. Math. 97, 101–123 (1975)

Download references


The authors would like to thank Craig Westerland for many helpful conversations. Without him we would not have started thinking about twisted \(\textit{BP} \langle {n} \rangle \)-cohomology and we would not have been led to the main result of this paper. The first author was supported by an ARC Discovery grant. The second author was partially supported by the DFG through SFB-1085, and thanks the Australian National University for hosting him while this research was conducted.

Author information

Correspondence to Vigleik Angeltveit.

Additional information

Communicated by Mark Behrens.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Angeltveit, V., Lind, J.A. Uniqueness of \(\textit{BP} \langle {n} \rangle \) . J. Homotopy Relat. Struct. 12, 17–30 (2017).

Download citation


  • Complex cobordism
  • Brown–Peterson spectrum
  • Adams spectral sequence