Algebraic & Geometric Topology

Stable and unstable operations in mod $p$ cohomology theories

Andrew Stacey and Sarah Whitehouse

Full-text: Open access


We consider operations between two multiplicative, complex orientable cohomology theories. Under suitable hypotheses, we construct a map from unstable to stable operations, left-inverse to the usual map from stable to unstable operations. In the main example, where the target theory is one of the Morava K–theories, this provides a simple and explicit description of a splitting arising from the Bousfield–Kuhn functor.

Article information

Algebr. Geom. Topol., Volume 8, Number 2 (2008), 1059-1091.

Received: 17 October 2006
Revised: 16 May 2008
Accepted: 19 May 2008
First available in Project Euclid: 20 December 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55S25: $K$-theory operations and generalized cohomology operations [See also 19D55, 19Lxx]
Secondary: 55P47: Infinite loop spaces

cohomology operations Morava K-theories


Stacey, Andrew; Whitehouse, Sarah. Stable and unstable operations in mod $p$ cohomology theories. Algebr. Geom. Topol. 8 (2008), no. 2, 1059--1091. doi:10.2140/agt.2008.8.1059.

Export citation


  • M Bendersky, The BP Hopf invariant, Amer. J. Math. 108 (1986) 1037–1058
  • J M Boardman, Stable operations in generalized cohomology, from: “Handbook of algebraic topology”, North-Holland, Amsterdam (1995) 585–686
  • J M Boardman, D C Johnson, W S Wilson, Unstable operations in generalized cohomology, from: “Handbook of algebraic topology”, North-Holland, Amsterdam (1995) 687–828
  • A K Bousfield, Uniqueness of infinite deloopings for $K$–theoretic spaces, Pacific J. Math. 129 (1987) 1–31
  • A K Bousfield, On the telescopic homotopy theory of spaces, Trans. Amer. Math. Soc. 353 (2001) 2391–2426
  • T Kashiwabara, N Strickland, P Turner, The Morava $K$–theory Hopf ring for $BP$, from: “Algebraic topology: new trends in localization and periodicity (Sant Feliu de Guí xols, 1994)”, Progr. Math. 136, Birkhäuser, Basel (1996) 209–222
  • N J Kuhn, Morava $K$–theories and infinite loop spaces, from: “Algebraic topology (Arcata, CA, 1986)”, Lecture Notes in Math. 1370, Springer, Berlin (1989) 243–257
  • N J Kuhn, Localization of André–Quillen–Goodwillie towers, and the periodic homology of infinite loopspaces, Adv. Math. 201 (2006) 318–378
  • D Quillen, On the formal group laws of unoriented and complex cobordism theory, Bull. Amer. Math. Soc. 75 (1969) 1293–1298
  • D C Ravenel, W S Wilson, The Hopf ring for complex cobordism, J. Pure Appl. Algebra 9 (1976/77) 241–280
  • C Rezk, The units of a ring spectrum and a logarithmic cohomology operation, J. Amer. Math. Soc. 19 (2006) 969–1014
  • W S Wilson, The Hopf ring for Morava $K$–theory, Publ. Res. Inst. Math. Sci. 20 (1984) 1025–1036