Brown–Peterson cohomology

Brown–Peterson cohomology
Name	Brown–Peterson cohomology
Field	Algebraic topology
Introduced	1966
Inventor	Edgar H. Brown Jr.; Franklin P. Peterson
Related	Complex cobordism, Morava K-theory, Adams–Novikov spectral sequence

Brown–Peterson cohomology is a family of extraordinary cohomology theories indexed by a prime p, developed to study periodic phenomena in stable homotopy theory. Originating from work by Edgar H. Brown Jr. and Franklin P. Peterson and refined through interactions with researchers such as Douglas C. Ravenel and Michael J. Hopkins, it occupies a central position connecting complex cobordism, Landweber exact functor theorem, and chromatic homotopy theory. The theory provides truncated variants of complex bordism that isolate p-typical information useful for computations with the Adams–Novikov spectral sequence and for understanding formal group laws arising in topology.

Introduction

Brown–Peterson cohomology was introduced to extract prime-specific invariants from Thom spectrum constructions linked to Milnor's and Novikov's work on cobordism. It is frequently denoted BP and plays a role analogous to Morava K-theory and Johnson–Wilson theory in the structural study of the stable homotopy groups of spheres, interacting with tools named for Adams, Novikov, and Hopkins. The formulation leverages algebraic structures studied by Hazewinkel and Cartier in the theory of formal groups and provides input to conjectures posed by Quillen and developed by Ravenel.

Construction and Definition

Brown–Peterson cohomology is constructed by localizing and summand-projecting the MU spectrum associated with complex cobordism at a fixed prime p, then splitting off the p-typical summand present in the Quillen formal group law decomposition. The resulting Brown–Peterson spectrum admits a coefficient ring BP_* = Z_(p)[v_1, v_2, ...] with generators v_i of graded degrees 2(p^i − 1), reflecting the p-typical Hazewinkel generators studied by Lazard and Dieudonné. The spectrum is typically built within the model categories used in modern stable homotopy theory, as developed by authors such as Hovey and Schwede, and it fits into the framework of structured ring spectra explored by Elmendorf, Mandell, and May.

Algebraic Structure and Operations

The coefficient algebra BP_* is a graded polynomial algebra over the localization Z_(p) with operations encoding the action of the Steenrod algebra's p-typical components and cooperations given by BP_*BP, which is a Hopf algebroid. This Hopf algebroid governs comodules that appear in the Adams–Novikov spectral sequence and relates to the stacks of one-dimensional formal groups studied by Deligne and Drinfeld. Chromatic operators such as the v_n-periodicity elements correspond to the height filtration central to work of Morava and Lubin–Tate, while Landweber exact functors and the Brown–Gitler style constructions give systematic ways to produce homology theories from BP through formal group law criteria pioneered by Landweber and applied by Frank Adams and Novikov.

Computations and Examples

Computations in Brown–Peterson cohomology often proceed via the Adams–Novikov spectral sequence, with E_2-terms expressed in Ext groups over BP_*BP. Classical calculations include BP_*(CP^n) and BP_*(BP), carried out using methods developed by Ravenel, Miller, and Wilson. Examples demonstrating the role of v_n-periodicity arise in analyses of the stable homotopy groups of spheres, where chromatic convergence results conjectured by Ravenel and proved in part by Devinatz, Hopkins, and Smith use BP-based techniques. Explicit low-dimensional computations employ the knowledge of BP_*BP comodules and connect to computations in Morava E-theory and Johnson–Wilson theory E(n).

Relations to Other Cohomology Theories

Brown–Peterson cohomology sits between complex cobordism MU and localized theories like Johnson–Wilson theory and Morava K-theory: BP is a summand of MU_(p) and admits Landweber exact maps to theories representing formal group laws of finite height such as Morava E-theory. The relationship to the Steenrod algebra and to the Adams spectral sequence provides bridges to ordinary singular cohomology and K-theory calculations, connecting classical results of Atiyah and Hirzebruch with modern chromatic perspectives developed by Hopkins, Ravenel, and Lurie. The comparative algebraic geometry viewpoint links BP to the moduli of formal groups investigated by Lubin, Tate, and Gross.

Applications and Impact

BP techniques have been pivotal in formulating and resolving structural conjectures in stable homotopy theory, including parts of the chromatic convergence theorem and the nilpotence and periodicity theorems influenced by work of Devinatz, Hopkins, and Smith. Brown–Peterson methods underpin computational breakthroughs by Ravenel and collaborators on periodic families in the stable homotopy groups of spheres and inform modern formulations in derived algebraic geometry as advanced by Lurie and Toën. Applications extend to the study of orientable genera studied by Hirzebruch and to connections with arithmetic geometry through the formal group frameworks developed by Serre and Grothendieck.

Category:Cohomology theories