Brown–Peterson cohomology

In mathematics, Brown–Peterson cohomology is a generalized cohomology theory introduced by Edgar H. Brown and Franklin P. Peterson (1966), depending on a choice of prime p. It is described in detail by Ravenel (2003, Chapter 4). Its representing spectrum is denoted by BP.

Complex cobordism and Quillen's idempotent

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.

For each prime p, Quillen showed there is a unique idempotent map of ring spectra ε from MUQ_(p) to itself, with the property that ε([CPⁿ]) is [CPⁿ] if n+1 is a power of p, and 0 otherwise. The spectrum BP is the image of this idempotent ε.

Structure of BP

The coefficient ring π_*(BP) is a polynomial algebra over Z_(p) on generators v_n of dimension 2(pⁿ − 1) for n ≥ 1.

BP_*(BP) is isomorphic to the polynomial ring π_*(BP)[t₁, t₂, ...] over π_*(BP) with generators t_i in BP_2(pⁱ−1)(BP) of degrees 2(pⁱ−1).

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.

See also

• List of cohomology theories#Brown–Peterson cohomology

References

• Adams, J. Frank (1974), Stable homotopy and generalised homology, University of Chicago Press, ISBN 978-0-226-00524-9 
• Brown, Edgar H., Jr.; Peterson, Franklin P. (1966), "A spectrum whose Z_p cohomology is the algebra of reduced p^th powers", Topology 5 (2): 149–154, doi:10.1016/0040-9383(66)90015-2, MR 0192494 .
• Quillen, Daniel (1969), "On the formal group laws of unoriented and complex cobordism theory", Bull. Amer. Math. Soc. 75 (6): 1293–1298, doi:10.1090/S0002-9904-1969-12401-8, MR 0253350 .
• Ravenel, Douglas C. (2003), Complex cobordism and stable homotopy groups of spheres (2nd ed.), AMS Chelsea, ISBN 0-8218-2967-X 
• Wilson, W. Stephen (1982), Brown-Peterson homology: an introduction and sampler, CBMS Regional Conference Series in Mathematics 48, Washington, D.C.: Conference Board of the Mathematical Sciences, ISBN 978-0-8219-1699-5, MR 655040