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 ${\text{BP}}$.

Complex cobordism and Quillen's idempotent

Brown–Peterson cohomology ${\text{BP}}$ 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 ${\text{BP}}$.

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 $\epsilon _{*}([CP^{n}])$ is [CP^$n$] if $n+1$ is a power of $p$, and $0$ otherwise. The spectrum ${\text{BP}}$ is the image of this idempotent ε.

Structure of ${\text{BP}}$

The coefficient ring $\pi _{*}({\text{BP}})$ is a polynomial algebra over $\mathbb {Z} _{(p)}$ on generators $v_{n}$ in degrees $2(p^{n}-1)$ for $n\geq 1$.

${\text{BP}}_{*}({\text{BP}})$ is isomorphic to the polynomial ring $\pi _{*}({\text{BP}})[t_{1},t_{2},\ldots ]$ over $\pi _{*}({\text{BP}})$ with generators $t_{i}$ in ${\text{BP}}_{2(p^{i}-1)}({\text{BP}})$ of degrees $2(p^{i}-1)$.

The cohomology of the Hopf algebroid $(\pi _{*}({\text{BP}}),{\text{BP}}_{*}({\text{BP}}))$ is the initial term of the Adams–Novikov spectral sequence for calculating $p$-local homotopy groups of spheres.

${\text{BP}}$ 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" (PDF), Bulletin of the American Mathematical Society, 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 978-0-8218-2967-7
• Wilson, W. Stephen (1982), Brown-Peterson homology: an introduction and sampler, CBMS Regional Conference Series in Mathematics, vol. 48, Washington, D.C.: Conference Board of the Mathematical Sciences, ISBN 978-0-8219-1699-5, MR 0655040