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 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 sum of suspensions of BP.

It is used as the initial term of the Adams-Novikov spectral sequence for calculating homotopy groups of spheres.

[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, MR0192494, DOI 10.1016/0040-9383(66)90015-2 .
• Quillen, Daniel (1969), “On the formal group laws of unoriented and complex cobordism theory”, Bull. Amer. Math. Soc. 75: 1293-1298, 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 edition ed.), AMS Chelsea, ISBN 0-8218-2967-X, <http://www.math.rochester.edu/people/faculty/doug/mu.html>