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

wedge product of suspensions

of BP.

For each prime p,

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 $\pi _{*}({\text{BP}})$ 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

.

BP is the universal example of a complex oriented cohomology theory whose associated formal group law is p-typical.

References

ISBN 978-0-226-00524-9

MR 0192494
.

MR 0253350
.

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,
MR 0655040

Complex cobordism and Quillen's idempotent

Structure of BP

See also

References