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 at a prime , hence the p-localization , decomposes as a direct sum. The direct summands are the Brown-Peterson spectra.
For each prime there is an unique commutative ring spectrum which is a retract of such that the map is multiplicative and such that
(…)
(…)
(…)
Due to (Brown-Peterson 66), recalled as (Ravenel, theorem, 4.1.12).
(…)
The formal group law of Brown-Peterson cohomology theory is universal for -local complex oriented cohomology theories in that is universal among -local, p-typical formal group laws.
(…)
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 -Steenrod algebra is described by the Adams-Quillen theorem.
The -component of the -term of the Adams-Novikov spectral sequence for the sphere spectrum, hence the one converging to the stable homotopy groups of spheres is
recalled e.g. as Ravenel, theorem 1.4.2
The spectrum can be constructed as a CW spectrum (cf. Priddy 1980) starting from the -local sphere spectrum by minimally attaching cells to to kill .
The original article is
Edgar Brown, F. P. Peterson, A spectrum whose cohomology is the algebra of reduced -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 -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.