Brown-Peterson spectrum

By the Pontryagin–Thom theorem, there is a ring spectrum $ MU $( cf. Spectrum of a ring) whose homotopy is isomorphic to the graded ring of bordism classes of closed smooth manifolds with a complex structure on their stable normal bundles (cf. also Cobordism). E.H. Brown and F.P. Peterson [a1] showed that, when localized at a prime $ p $, the spectrum $ MU $ is homotopy equivalent to the wedge of various suspensions (cf. also Suspension) of a ring spectrum $ BP $, the Brown–Peterson spectrum. The homotopy of this spectrum is the polynomial algebra

$$ \pi _ {*} BP = \mathbf Z _ {( p ) } [ v _ {1} \dots v _ {n} , \dots ] , $$

where the degree of $ v _ {n} $ is $ 2 ( p ^ {n} - 1 ) $. As a module over the Steenrod algebra,

$$ H _ {*} ( BP; \mathbf Z/p ) \simeq \left \{ \begin{array}{l} {\mathbf Z/2 [ \xi _ {1} ^ {2} \dots \xi _ {n} ^ {2} , \dots ] , \ p = 2, } \\ {\mathbf Z/p [ \xi _ {1} \dots \xi _ {n} , \dots ] , \ p \textrm{ odd } . } \end{array} \right . $$

Four properties of $ BP $ have made it one of the most useful spectra in homotopy theory. First, D. Quillen [a5] determined the structure of its ring of operations. Second, A. Liulevicius [a3] and M. Hazewinkel [a2] constructed polynomial generators of $ \pi _ {*} BP $ with good properties. Third, the Baas–Sullivan construction can be used to construct simple spectra from $ BP $ with very nice properties. The most notable of these spectra are the Morava $ K $- theories $ K ( n ) $, which are central in the statement of the periodicity theorem. (See [a7] for an account of the nilpotence and periodicity theorems.) Fourth, S.P. Novikov [a4] constructed the Adams–Novikov spectral sequence, which uses knowledge of the Brown–Peterson homology of a spectrum $ X $ to compute the homotopy of $ X $. (See [a6] for a survey of how the Adams–Novikov spectral sequence gives information on the stable homotopy groups of spheres.)

An introduction to the study of $ BP $ is given in [a8].

References

[a1]	E.H. Brown, F.P. Peterson, "A spectrum whose $\ZZ_p$-homology is the algebra of reduced $p$-th powers" Topology , 5 (1966) pp. 149–154
[a2]	M. Hazewinkel, "Constructing formal groups III. Applications to complex cobordism and Brown–Peterson cohomology" J. Pure Appl. Algebra , 10 (1977/78) pp. 1–18
[a3]	A. Liulevicius, "On the algebra $BP^\star(BP)$" , Lecture Notes in Mathematics , 249 , Springer (1971) pp. 47–53
[a4]	S.P. Novikov, "The methods of algebraic topology from the viewpoint of cobordism theories" Math. USSR Izv. (1967) pp. 827–913 Izv. Akad. Nauk SSSR Ser. Mat. , 31 (1967) pp. 855–951
[a5]	D. Quillen, "On the formal group laws of unoriented and complex cobordism theory" Bull. Amer. Math. Soc. , 75 (1969) pp. 1293–1298
[a6]	D.C. Ravenel, "Complex cobordism and stable homotopy groups of spheres" , Pure and Applied Mathematics , 121 , Acad. Press (1986)
[a7]	D.C. Ravenel, "Nilpotence and periodicity in stable homotopy theory" , Annals of Math. Stud. , 128 , Princeton Univ. Press (1992)
[a8]	W.S. Wilson, "Brown–Peterson homology, an introduction and sampler" , Regional Conf. Ser. Math. , 48 , Amer. Math. Soc. (1982)