group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
symmetric monoidal (∞,1)-category of spectra
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.
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
(…)
(…)
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Due to (Brown-Peterson 66), recalled as (Ravenel, theorem, 4.1.12).
(…)
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.
(…)
$B P$ is related to p-typical formal groups as MU is to formal groups.
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.
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
recalled e.g. as Ravenel, theorem 1.4.2
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 $\pi_{2n+1}(X_n)$.
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
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
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)
Last revised on November 30, 2020 at 17:55:30. See the history of this page for a list of all contributions to it.