Brown–Peterson spectrum

Brown–Peterson spectrum
Name	Brown–Peterson spectrum
Discipline	Algebraic topology
Introduced	1960s
Founder	Edgar H. Brown Jr.; Franklin P. Peterson
Related	Complex cobordism; Morava K-theory; Adams–Novikov spectral sequence

Brown–Peterson spectrum The Brown–Peterson spectrum is a family of spectra in algebraic topology introduced by Edgar H. Brown Jr. and Franklin P. Peterson that isolates p‑primary information in complex cobordism and provides a computationally tractable object for stable homotopy theory. It plays a central role in the work of J. Frank Adams, Sergei Novikov, John Milnor, Michael Atiyah, and Haynes Miller through its connections to the Adams spectral sequence, Adams–Novikov spectral sequence, and the study of formal group laws associated to Lazard's theorem.

Introduction

The construction of the Brown–Peterson spectrum arose in the context of the classification of complex-oriented cohomology theories and the study of Milnor's structure theorem for Steenrod algebra operations at a fixed prime p. Early developments involved contributions by René Thom, Daniel Quillen, William Browder, Douglas Ravenel, and Frank Adams who framed chromatic phenomena in terms of the Brown–Peterson theory. The spectrum refines ideas from complex cobordism and isolates distinguished generators related to Hazewinkel and Araki generators for the p‑typical formal group law studied by Lazard and Quillen.

Construction and Homotopy Properties

Brown–Peterson theory is defined at a prime p by taking a summand of the localized complex cobordism spectrum MU_{(p)}, following methods related to idempotents studied by Milnor and Novikov. The resulting spectrum BP_p has homotopy groups BP_* = Z_{(p)}[v_1, v_2, v_3, ...] where the degrees of the Hazewinkel or Araki generators v_n are 2(p^n−1). These homotopy groups interact with the Steenrod algebra at p through the dual Steenrod algebra calculations of Milnor and subsequent structural results by Adams, Priddy, Margolis, and Palmieri. BP admits structured ring spectrum enhancements studied by Elmendorf, Kriz, Mandell, and May, enabling multiplicative and module-theoretic techniques analogous to work by Boardman and Bousfield.

Relationship to Complex Cobordism and MU

The relationship between BP and MU is mediated by the p‑typification map studied in Quillen’s work on complex cobordism and formal groups; MU splits after p‑localization into pieces whose chromatic information is concentrated in BP. Quillen’s theorem on formal group laws links MU_* to the Lazard ring, while projection onto BP isolates the p‑typical part considered by Hazewinkel and Araki. This relationship is foundational to comparisons performed by Novikov in the Adams–Novikov spectral sequence and to calculations by Ravenel and Wilson that exploit BP_*BP comodules and the Landweber exact functor theorem associated to Landweber and Stong.

Chromatic Perspective and Morava K‑theories

In the chromatic viewpoint developed by Ravenel, Hopkins, Smith, and Nilpotence theorem contributors, BP provides the base for stratification by height via its vn‑periodic families. The sequence of periodicity generators v_n in BP_* detects layers corresponding to Morava theories: the Morava K‑theories K(n) of Jack Morava and the Morava E‑theories E_n associated to Lubin–Tate deformation theory and studied by Goerss, Hopkins, Miller, and Lurie. The nilpotence and periodicity results of Devinatz, Hopkins, and Smith use BP-based telescopes and localizations central to the classification of thick subcategories by Balmer and the chromatic fracture squares employed in work by Hovey and Sadofsky.

Computations and Important Results

Key computations include the structure of BP_*BP as a Hopf algebroid, computed using methods of Milnor and Novikov, and exploited in Adams–Novikov spectral sequence calculations by Adams, Novikov, Toda, Mahowald, and Toda bracket analyses by Cohen. Important results include Landweber exactness criteria applied to modules over BP_* by Landweber and the construction of complex orientations via Brown–Peterson cohomology used by Wilson and Johnson. Periodicity theorems identifying vn‑periodic families were established in work by Ravenel and refined in the chromatic convergence work of Hopkins, Ravenel, and Hovey. Calculations of the homotopy of BP-local spheres, and the role of BP in the computation of stable homotopy groups of spheres, were advanced by Adams, Barratt, Mahowald, and Toda using the Adams–Novikov machine.

Applications and Examples

Applications of BP include construction of complex-oriented cohomology theories via the Landweber exact functor theorem used by Landweber and Stong, analyses of periodic families in the stable homotopy groups of spheres pursued by Ravenel and Mahowald, and connections to elliptic cohomology and topological modular forms developed by Witten, Hopkins, Goerss, and Miller. Examples where BP and its localizations appear are computations of cobordism rings associated to manifolds studied by Thom and Milnor, the study of formal group law invariants in work by Quillen and Hazewinkel, and chromatic redshift phenomena considered by Clausen, Mathew, and Morrow in recent algebraic K‑theory and motivic contexts.

Category:Spectra