nLab Brown-Peterson spectrum

Idea

The localization of complex cobordism cohomology theory $MU$ at a prime $p$, hence the p-localization $MU_{(p)}$, decomposes as a direct sum. The direct summands are the Brown-Peterson spectra.

Definition

Theorem

For each prime $p$ there is an unique commutative ring spectrum $B P$ which is a retract of $M U_{(p)}$ such that the map $MU_{(p)} \to B P$ is multiplicative and such that

1. (…)

2. (…)

3. (…)

Due to (Brown-Peterson 66), recalled as (Ravenel, theorem, 4.1.12).

Properties

Universal characterization

(…)

The formal group law of Brown-Peterson cohomology theory is universal for $p$-local complex oriented cohomology theories in that $\mathbb{G}_{B P}$ is universal among $p$-local, p-typical formal group laws.

(…)

Relation to $p$-typical formal groups

$B P$ is related to p-typical formal groups as MU is to formal groups.

Hopf algebroid structure on dual BP-Steenrod algebra

The structure of Hopf algebroid over a commutative base on the dual $BP$-Steenrod algebra $BP_\bullet(BP)$ is described by the Adams-Quillen theorem.

Relation to Adams-Novikov spectral sequence

The $p$-component of the $E^2$-term of the Adams-Novikov spectral sequence for the sphere spectrum, hence the one converging to the stable homotopy groups of spheres $\pi_\ast(\mathbb{S})$ is

$$Ext_{BP_\ast(BP)}(BP_\ast, BP_\ast) \,.$$

recalled e.g. as Ravenel, theorem 1.4.2

As a CW spectrum

The spectrum $B P$ can be constructed as a CW spectrum (cf. Priddy 1980) starting from the $p$-local sphere spectrum $S^0 = X_0$ by minimally attaching cells to $X_n$ to kill $\pi_{2n+1}(X_n)$.

Related concepts

• Morava E-theory

• BPR-theory

References

The original article is

• Edgar Brown, F. P. Peterson, A spectrum whose $\mathbb{Z}/p$ cohomology is the algebra of reduced $p$-th powers, Topology 5 (1966) 149.

• Frank Adams, part II.16 of Stable homotopy and generalised homology,1974

An alternate construction was noted by Priddy

• Stewart Priddy, A cellular construction of BP and other irreducible spectra, Math. Z. 173 (1980), pp 29-34.

A textbook accounts:

• Doug Ravenel, section 4 (pdf) in: Complex cobordism and stable homotopy groups of spheres (web)

• Yuli Rudyak, Section IX.6 in: On Thom Spectra, Orientability, and Cobordism, Springer 1998 (doi:10.1007/978-3-540-77751-9)

The truncated version is discussed in

• Tyler Lawson, Niko Naumann, Truncated Brown-Peterson spectra (2012) (pdf)

On the Hopf invariant and e-invariant in BP-theory:

• Yasumasa Hirashima, On the $BP_\ast$-Hopf invariant, Osaka J. Math., Volume 12, Number 1 (1975), 187-196 (euclid:ojm/1200757733)

• Martin Bendersky, The BP Hopf Invariant, American Journal of Mathematics, Vol. 108, No. 5 (Oct., 1986) (jstor:2374595)