Brown–Peterson cohomology

Short description: Mathematical generalized cohomology theory

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 ε([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 ${\mathbb{\mathbb{Z}}}_{(p)}$ on generators $v_n$ in degrees $2(p^n-1)$ for $n\ge 1$.

${\text{BP}}_{*}(\text{BP})$ is isomorphic to the polynomial ring ${\pi }_{*}(\text{BP})[t_1,t_2,\dots ]$ 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

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