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 Douglas 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, Daniel 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 [math]\displaystyle{ \pi_*(\text{BP}) }[/math] is a polynomial algebra over [math]\displaystyle{ \Z_{(p)} }[/math] on generators [math]\displaystyle{ v_n }[/math] in degrees [math]\displaystyle{ 2(p^n-1) }[/math] for [math]\displaystyle{ n\ge 1 }[/math].

[math]\displaystyle{ \text{BP}_*(\text{BP}) }[/math] is isomorphic to the polynomial ring [math]\displaystyle{ \pi_*(\text{BP})[t_1, t_2, \ldots] }[/math] over [math]\displaystyle{ \pi_*(\text{BP}) }[/math] with generators [math]\displaystyle{ t_i }[/math] in [math]\displaystyle{ \text{BP}_{2 (p^i-1)}(\text{BP}) }[/math] of degrees [math]\displaystyle{ 2 (p^i-1) }[/math].

The cohomology of the Hopf algebroid [math]\displaystyle{ (\pi_*(\text{BP}), \text{BP}_*(\text{BP})) }[/math] 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

References

• Adams, J. Frank (1974), Stable homotopy and generalised homology, University of Chicago Press, ISBN 978-0-226-00524-9, https://books.google.com/books?id=6vG13YQcPnYC 
• 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 .
• Quillen, Daniel (1969), "On the formal group laws of unoriented and complex cobordism theory", Bulletin of the American Mathematical Society 75 (6): 1293–1298, doi:10.1090/S0002-9904-1969-12401-8, https://www.ams.org/journals/bull/1969-75-06/S0002-9904-1969-12401-8/S0002-9904-1969-12401-8.pdf .
• Ravenel, Douglas C. (2003), Complex cobordism and stable homotopy groups of spheres (2nd ed.), AMS Chelsea, ISBN 978-0-8218-2967-7, http://www.math.rochester.edu/people/faculty/doug/mu.html 
• 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, https://books.google.com/books?id=0KgfHsGe4hgC