View source for Brown-Peterson spectrum

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By the Pontryagin–Thom theorem, there is a ring spectrum  $  MU $(
cf. [[Spectrum of a ring|Spectrum of a ring]]) whose [[Homotopy|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|Cobordism]]). E.H. Brown and F.P. Peterson [[#References|[a1]]] showed that, when localized at a prime  $  p $, 
the spectrum  $  MU $
is homotopy equivalent to the wedge of various suspensions (cf. also [[Suspension|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|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 [[#References|[a5]]] determined the structure of its ring of operations. Second, A. Liulevicius [[#References|[a3]]] and M. Hazewinkel [[#References|[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 [[#References|[a7]]] for an account of the nilpotence and periodicity theorems.) Fourth, S.P. Novikov [[#References|[a4]]] constructed the Adams–Novikov [[Spectral sequence|spectral sequence]], which uses knowledge of the Brown–Peterson homology of a spectrum  $  X $
to compute the homotopy of  $  X $. 
(See [[#References|[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 [[#References|[a8]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M. Hazewinkel,   "Constructing formal groups III. Applications to complex cobordism and Brown–Peterson cohomology"  ''J. Pure Appl. Algebra'' , '''10'''  (1977/78)  pp. 1–18</TD></TR>
<TR><TD valign="top">[a3]</TD> <TD valign="top">  A. Liulevicius,   "On the algebra $BP^\star(BP)$" , ''Lecture Notes in Mathematics'' , '''249''' , Springer  (1971)  pp. 47–53</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  D. Quillen,   "On the formal group laws of unoriented and complex cobordism theory"  ''Bull. Amer. Math. Soc.'' , '''75'''  (1969)  pp. 1293–1298</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  D.C. Ravenel,   "Complex cobordism and stable homotopy groups of spheres" , ''Pure and Applied Mathematics'' , '''121''' , Acad. Press  (1986)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  D.C. Ravenel,   "Nilpotence and periodicity in stable homotopy theory" , ''Annals of Math. Stud.'' , '''128''' , Princeton Univ. Press  (1992)</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  W.S. Wilson,   "Brown–Peterson homology, an introduction and sampler" , ''Regional Conf. Ser. Math.'' , '''48''' , Amer. Math. Soc.  (1982)</TD></TR></table>