Brown–Peterson cohomology

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In mathematics, Brown–Peterson cohomology is a generalized cohomology theory introduced by Brown & Peterson (1966), depending on a choice of prime p. It is described in detail by Ravenel (2003, Chapter 4).

Its representing spectrum is usually denoted by BP. Its coefficient ring π_*(BP) is a polynomial algebra over Z_(p) on generators v_n of dimension 2(pⁿ − 1) for n ≥ 1. Brown–Peterson cohomology BP is a summand of MU_p, which is complex cobordism MU localized at a prime p. In fact MU_(p) is a wedge product of suspensions of BP.

The cohomology of the Hopf algebroid (π_*(BP), BP_*(BP)) is the initial term of the Adams-Novikov spectral sequence for calculating p-local homotopy groups of spheres.

BP is the universal example of a complex oriented cohomology theory whose associated formal group law is p-typical.

[edit] See also

• List of cohomology theories#Brown–Peterson cohomology

[edit] References

• Brown, Edgar H., Jr.; Peterson, Franklin P. (1966), "A spectrum whose Z_p cohomology is the algebra of reduced p^th powers.", Topology 5: 149--154, doi:10.1016/0040-9383(66)90015-2, MR0192494 .
• Quillen, Daniel (1969), "On the formal group laws of unoriented and complex cobordism theory" (^{[dead link]} – ^{Scholar search}), Bull. Amer. Math. Soc. 75: 1293–1298, doi:10.1090/S0002-9904-1969-12401-8, MR0253350, http://www.ams.org/bull/1969-75-06/S0002-9904-1969-12398-0/S0002-9904-1969-12398-0.pdf .
• Ravenel, Douglas C. (2003), Complex cobordism and stable homotopy groups of spheres (2nd ed.), AMS Chelsea, ISBN 0-8218-2967-X, http://www.math.rochester.edu/people/faculty/doug/mu.html