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 vn of dimension 2(pn − 1) for n ≥ 1. Brown–Peterson cohomology BP is a summand of MUp, 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.
See also
References
- Brown, Edgar H., Jr. & Peterson, Franklin P. (1966), "A spectrum whose Zp cohomology is the algebra of reduced pth 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>
