View source for Cobordism

<!--
c0227801.png
$#A+1 = 540 n = 4
$#C+1 = 540 : ~/encyclopedia/old_files/data/C022/C.0202780 Cobordism,
Automatically converted into TeX, above some diagnostics.
Please remove this comment and the {{TEX|auto}} line below,
if TeX found to be correct.
-->

{{TEX|auto}}
{{TEX|done}}

''cobordism theory''

A generalized cohomology theory determined by spectra of Thom spaces and related to various structures in the stable tangent or normal bundle to a manifold. Cobordism theory is dual (in the sense of [[S-duality| $  S $-
duality]]) to the theory of [[Bordism|bordism]].

The simplest example of cobordism is orthogonal or non-oriented cobordism. Let  $  O _ {r} $
by the group of orthogonal transformations of the Euclidean space  $  \mathbf R  ^ {n} $, 
and  $  {BO } _ {r} $
its [[Classifying space|classifying space]]. The standard imbedding  $  O _ {r} \rightarrow O _ {r+} 1 $
defines a mapping  $  j _ {r} :  BO _ {r} \rightarrow {BO } _ {r+} 1 $
taking the universal fibre bundle  $  \gamma _ {r+} 1 $
over  $  {BO } _ {r+} 1 $
into the bundle  $  \gamma _ {r} \oplus \theta $, 
where  $  \theta $
is the one-dimensional trivial bundle over  $  {BO } _ {r} $. 
If  $  {TBO } _ {r} $
is the [[Thom space|Thom space]] of  $  \gamma _ {r} $, 
then one obtains a mapping  $  s _ {r} :  STBO _ {r} \rightarrow {TBO } _ {r+} 1 $
induced by  $  j _ {r} $, 
where  $  S $
is [[Suspension|suspension]]. The sequence  $  \{ TBO _ {r} , s _ {r} \} $
forms a [[Spectrum of spaces|spectrum of spaces]] and therefore defines a cohomology theory, called the theory of orthogonal cobordism or non-oriented cobordism or  $  O $-
cobordism; it is denoted by  $  O  ^ {*} $. 
The group  $  O  ^ {n} ( X , A ) $
of  $  n $-
dimensional  $  O $-
cobordism of the pair  $  ( X , A ) $
is defined as

$$ 
\lim\limits _
{i \rightarrow \infty } \ 
[ S  ^ {i} ( X / A ) ,\ 
T {BO } _ {i+} n ] ,
$$

where  $  [ P , Q ] $
is the set of homotopy classes of mappings from  $  P $
into  $  Q $. 
Here  $  O  ^ {n} ( x) = O  ^ {n} ( X , \emptyset ) $, 
$  \emptyset $
is the empty set, and by  $  X / \emptyset = X  ^ {+} $
one means the disjoint union of  $  X $
and a point. The group  $  O  ^ {n} ( X , x _ {0} ) $, 
where  $  x _ {0} \in X $, 
is called the reduced group of  $  n $-
dimensional  $  O $-
cobordism  $  \widetilde{O}  {}  ^ {n} ( X) $
of  $  X $. 
The generalized homology theory dual to the  $  O $-
cobordism theory is called  $  O $-
bordism theory. The groups  $  O _ {n} ( X , A ) $
of  $  n $-
dimensional bordism of the pair  $  ( X , A ) $
are defined as

$$ 
\lim\limits _
{i \rightarrow \infty } \ 
\pi _ {i+} n ( ( X / A )
\wedge T {BO } _ {i} ) .
$$

The groups of  $  n $-
dimensional  $  O $-
bordism of a point are denoted by  $  \Omega _ {O}  ^ {n} $
and the  $  n $-
dimensional  $  O $-
bordism of a point by  $  \Omega _ {n}  ^ {O} $; 
the latter can be described purely geometrically. Furthermore,  $  \Omega _ {O}  ^ {-} n \approx \Omega _ {n}  ^ {O} \approx \pi _ {n+} N ( T {BO } _ {N} ) $, 
$  N \gg n $, 
so that it can be interpreted both as a cobordism group and a bordism group (see [[Bordism|bordism]], where it is denoted by  $  \mathfrak N _ {n} $). 
The total coefficient group of  $  O $-
cobordism theory, the graded group  $  \Omega _ {O} = \oplus _ {- \infty }  ^ {+ \infty } \Omega _ {O}  ^ {n} $, 
is a ring: multiplication is induced by the Cartesian product of manifolds. Furthermore, for any finite  $   \mathop{\rm CW} $-
complex  $  X $
the group  $  O ( X) = \oplus _ {n= - \infty }  ^ {+ \infty } O  ^ {n} ( X) $
is a natural ring with respect to  $  X $
since the mapping  $  {BO } _ {m} \times {BO } _ {n} \rightarrow {BO } _ {m+} n $
induced by the imbedding  $  O _ {m} \times O _ {n} \rightarrow O _ {m+} n $
defines a mapping  $  T {BO } _ {m} \wedge T {BO } _ {n} \rightarrow T {BO } _ {m+} n $, 
so that  $  \{ T {BO } _ {r} \} $
is a multiplicative spectrum of spaces.

The general situation is described as follows. By a structural series  $  ( B , \phi ) $
one means a sequence of bundles  $  \phi _ {r} :  B _ {r} \rightarrow {BO } _ {r} $
and mappings  $  i _ {r} :  B _ {r} \rightarrow B _ {r+} 1 $
such that  $  \phi _ {r+} 1 \circ i _ {r} = j _ {r} \circ \phi _ {r} $. 
The mapping  $  \phi _ {r} $
defines a vector bundle  $  \xi _ {r} = \phi  ^ {*} \gamma _ {r} $
over  $  B _ {r} $, 
where  $  i _ {r}  ^ {*} \xi _ {r+} 1 = \xi _ {r} + \phi _ {r}  ^ {*} \theta $. 
Let  $  TB _ {r} $
be the Thom space of the bundle  $  \xi _ {r} $; 
the above equality defines a mapping  $  s _ {r} :  STB _ {r} \rightarrow TB _ {r+} 1 $
such that the sequence  $  T ( B , \phi ) = \{ TB _ {r} , s _ {r} \} $
is a spectrum of spaces, hence defines a cohomology theory. It is called  $  ( B , \phi ) $-
cobordism theory and is denoted by  $  ( B , \phi )  ^ {*} $. 
Thus,

$$ 
( B \phi )  ^ {i}
( X , A )  = \ 
\lim\limits _
{N \rightarrow \infty } \ 
[ S  ^ {N} ( X / A ) ,\ 
T B _ {i+} N ] .
$$

The coefficient group of the  $  ( B , \phi ) $-
cobordism theory is denoted by  $  \Omega _ {( B , \phi ) }  $. 
Here,  $  \Omega _ {i} ^ {( B , \phi ) } = \Omega _ {( B , \phi ) }   ^ {-} i = \pi _ {i+} N ( T B _ {N} ) $, 
$  N \gg i $, 
where  $  \Omega _ {i} ^ {( B , \phi ) } $
is the coefficient group of the dual  $  ( B , \phi ) $-
bordism theory, which admits of a geometric definition using the concept of a so-called [[B-Phi-structure| $  ( B , \phi ) $-
structure]]:  $  ( B , \phi ) $-
bordancy is defined and the elements of  $  \Omega ^ {( B , \phi ) } $
are interpreted as classes of  $  ( B , \phi ) $-
bordant manifolds.

The first examples of cobordism theory arose from series of linear groups. For example, the series of orthogonal groups  $  \{ O _ {r} \} $
defines the structural series  $  \{ B _ {r} , \phi _ {r} \} $, 
where  $  B _ {r} = {BO } _ {r} $, 
$  \phi _ {r} =  \mathop{\rm id} $. 
The series  $  \{ SO _ {r} \} $
defines the structural series  $  \{ B _ {r} , \phi _ {r} \} $, 
where  $  B _ {r} = B SO _ {r} $
and  $  \phi _ {r} :  B SO _ {r} \rightarrow {BO } _ {r} $
is the universal two-sheeted covering corresponding to the inclusion  $  SO _ {r} \subset  O _ {r} $. 
The corresponding cobordism theory is called the theory of oriented cobordism; it is denoted by  $  SO  ^ {*} $. 
The series of unitary groups  $  \{ U _ {r} \} $
defines a theory of unitary or complex cobordism, quasi-complex cobordism, almost-complex cobordism; it is denoted by  $  U  ^ {*} $. 
Here the series  $  \{ B , \phi \} $
is constructed in the following way:  $  B _ {2r} = B _ {2r+} 1 = {BU } _ {r} $
is classifying space of  $  U _ {r} $
and the  $  \phi _ {r} $, 
$  \phi _ {2r+} 1 $
are the mappings of the classifying spaces  $  BU _ {r} \rightarrow {BO } _ {2r} $
and  $  {BU } _ {r} \rightarrow {BO } _ {2r} \rightarrow BO _ {2r+} 1 $, 
respectively, induced by the natural imbeddings  $  U _ {r} \subset  O _ {2r} \subset  O _ {2r+} 1 $. 
The series of symplectic groups  $  \{  \mathop{\rm Sp} _ {r} \} $
defines a theory of symplectic cobordism,  $   \mathop{\rm Sp}  ^ {*} $, 
where  $  B _ {4r} = B _ {4r+} 1 = B _ {4r+} 2 = B _ {4r+} 3 = B  \mathop{\rm Sp} _ {r} $, 
and the  $  \phi _ {r} $
are constructed in the same way as for the unitary case. There are also cobordism theories corresponding to the series of groups  $  \{  \mathop{\rm Spin} _ {r} \} $, 
$  \{ SU _ {r} \} $, 
etc. Finally, the series of identity groups  $  \{ E _ {r} \} $, 
where  $  \phi _ {r} :  B _ {r} \rightarrow {BO } _ {r} $
is a fibre bundle with contractible  $  B _ {r} $, 
defines a cobordism theory which is the same as the theory of stable cohomotopy groups, and therefore the dual bordism theory is isomorphic to the theory of stable homotopy groups,  $  E _ {i} ( X) \approx \pi _ {i+} N ( S  ^ {N} X ) $, 
$  N \gg i $. 
An  $  E $-
manifold is said to be framed (trivialized) since the  $  E $-
structure is precisely a frame (trivialization) of the stable normal bundle.  $  E $-
cobordism theory is called trivialized or framed cobordism theory, its  $  i $-
dimensional coefficient group being denoted by  $  \Omega _ { \mathop{\rm fr}  }  ^ {i} $, 
so that  $  \Omega _ { \mathop{\rm fr}  }  ^ {-} i = \Omega _ {i} ^ { \mathop{\rm fr} } = \pi _ {i+} N ( S  ^ {N} ) $. 
This is the first example of a cobordism; it was due to L.S. Pontryagin, who interpreted the stable homotopy groups of the sphere as (geometrically defined) groups of framed cobordism of a point of  $  \Omega _ { \mathop{\rm fr}  }  ^ {i} $, 
with the aim of computing the group  $  \pi _ {i+} N ( S  ^ {N} ) $.

All these cobordism theories arising from series of linear groups are multiplicative, and therefore for any finite  $   \mathop{\rm CW} $-
complex  $  X $, 
the total (graded) cobordism group is a ring. For example, for the series of groups  $  \{ U _ {r} \} $
there is an imbedding  $  U _ {m} \times U _ {n} \rightarrow U _ {m+} n $
inducing a mapping

$$ 
{BU } _ {m} \times BU _ {n}  \rightarrow  BU _ {m+} n ,
$$

and therefore a mapping  $  T {BU } _ {m} \wedge T {BU } _ {n} \rightarrow T {BU } _ {m+} n $. 
The spectrum  $  \{ M _ {r} \} $
representing the theory  $  U  ^ {*} $
has the form  $  M _ {2r} = T {BU } _ {r} $, 
$  M _ {2r+} 1 = S T BU _ {r} $, 
hence there exist mappings  $  M _ {r} \wedge M _ {s} \rightarrow M _ {r+} s $
so that the spectrum of spaces  $  \{ M _ {r} \} $
is multiplicative.

The development of cobordism theory started from the geometric definition and calculation of the groups  $  \Omega _ {E} $, 
$  \Omega _ {O }  $, 
$  \Omega _ {SO }  $. 
An important role was played by Pontryagin's theorem stating that  $  O $-
bordant manifolds have the same Stiefel number. The study of cobordism theory was advanced by R. Thom. He introduced the spaces  $  T {BO } _ {N} $, 
$  TB SO _ {N} $
and proved the isomorphism  $  \pi _ {i+} N ( T {BO } _ {N} ) \approx \Omega _ {SO}  ^ {-} i $, 
enabling one to bring into the calculation of the cobordism rings some of the methods of homotopic topology. Thom's constructions stimulated the introduction of  $  T {BU } _ {n} $, 
$  T B  \mathop{\rm Sp} _ {n} $, 
etc., and the corresponding cobordisms. The fundamental problem of the first stage of the development of cobordism theory was the calculation of the cobordism rings of a point.

In the study of the cobordism of a point a big part is played by the characteristic classes: Chern classes for  $  \Omega _ {U} $, 
Stiefel classes for  $  \Omega _ {O} $, 
Pontryagin and Stiefel classes for  $  \Omega _ {SO} $(
cf. [[Characteristic class|Characteristic class]]; [[Chern class|Chern class]]; [[Pontryagin class|Pontryagin class]]). In general, given any structural series  $  ( B , \phi ) $
and any multiplicative cohomology theory  $  h  ^ {*} $
in which all bundles  $  \xi _ {r} $
over  $  B _ {r} $
are orientable, one can define the characteristic classes as elements of the group  $  h  ^ {*} ( B) $, 
where  $  B = \lim\limits  ( B _ {r} , j _ {r} ) $. 
Furthermore, the corresponding characteristic numbers, which are elements of the rings  $  h  ^ {*} (  \mathop{\rm pt} ) $
are invariant with respect to  $  ( B , \phi ) $-
bordancy. Let  $  \omega = ( i _ {1} \dots i _ {k} ) $
be a partition of  $  n $
and let  $  S _  \omega  $
be the symmetric function of  $  n $
variables corresponding to  $  \omega $. 
The characteristic class  $  S _  \omega  ( c _ {1} \dots c _ {n} ) $(
see [[Chern class|Chern class]]) is denoted by  $  S _  \omega   ^ {c} $. 
The analogous constructions for the Pontryagin and Stiefel classes are denoted by  $  S _  \omega   ^ {p} $
and  $  S _  \omega   ^ {s} $, 
respectively.

1) Unitary cobordism. The ring  $  \Omega _ {U} $
is the free graded polynomial algebra in a countable number of homogeneous generators

$$ 
\Omega _ {U}  =  \mathbf Z [ x _ {1} \dots x _ {n} ,\dots ] ,\   \mathop{\rm deg} \ 
x _ {i}  =  - 2 i .
$$

The set  $  \{ x _ {n} \} $, 
$   \mathop{\rm deg}  x _ {n} = - 2 n $, 
is a system of polynomial generators if and only if

$$ 
S _ {(} n)  ^ {c} ( x _ {n} )  = \ 
\left \{ 
\begin{array}{lll}
\pm  1 ,  &n \neq p  ^ {r} - 1  &\textrm{ for  any  "prime"  }  p  \textrm{ and  integer  }  r ,  \\
\pm  p ,  &n = p  ^ {r} - 1  &\textrm{ for  some  "prime"   }  p  \textrm{ and  integer  }  r ,  \\
\end{array}

 \right .$$

where  $  ( n) $
is the partition of  $  n $
consisting of a single term. One of the systems of polynomial generators of  $  \Omega _ {U} $
can be described as follows. Let  $  \mathbf C P  ^ {n} $
be  $  n $-
dimensional complex projective space. The complex algebraic hypersurface of bidegree  $  ( 1 , 1 ) $
in  $  \mathbf C P  ^ {i} \times \mathbf C P  ^ {j} $
is a complex manifold. Its unitary cobordism class is denoted by  $  H _ {i,j} $, 
$   \mathop{\rm dim} _ {\mathbf R}  H _ {i,j} = 2 ( i + j - 1 ) $. 
It turns out that

$$ 
S _ {i+} j- 1 ( H _ {i,j} )  = \ 
\left ( \begin{array}{c}
i+ j \\
 i 
\end{array}
 \right ) ,
$$

so that an appropriate linear integer combination of elements of  $  H _ {i,j} $
defines a generator of  $  \Omega _ {U} $
of degree  $  2 ( 1 - j - i ) $.

Since  $  \Omega _ {U} $
is torsion-free and  $  H  ^ {*} ( {BU } ;  \mathbf Z ) = \mathbf Z ( c _ {1} \dots c _ {n} ,\dots ) $, 
where the  $  c _ {n} $
are the Chern classes,  $   \mathop{\rm deg}  c _ {n} = 2 n $(
cf. [[Chern class|Chern class]]), it follows that the Chern numbers (cf. [[Chern number|Chern number]]) completely determine the unitary cobordism class of an almost-complex manifold.

Let  $  n $
be a positive integer and let  $  ( i _ {1} \dots i _ {k} ) $, 
$  i _ {s} > 0 $, 
$  \sum i _ {s} = n $, 
be a partition of it. There corresponds to each  $  2n $-
dimensional (real dimension) almost-complex manifold  $  M $
a set  $  \{ a _ {i _ {1}  \dots i _ {k} } \} = \{ c _ {i _ {1}  } \dots c _ {i _ {k}  } ( M) \} $
of integers, where the multi-index  $  i _ {1} \dots i _ {k} $
runs through all the partitions of  $  n $. 
A set of such integers  $  \{ b _ {i _ {1}  } \dots b _ {i _ {k}  } \} $
is realized as the set of Chern numbers of some almost-complex manifold in the following situation. Let  $  S _  \omega   ^ {c} ( e) \in H  ^ {**} ( {BU } ;  \mathbf Q ) $
be the [[Characteristic class|characteristic class]] given by replacing the Wu generators  $  x _ {i} $
in the representation of  $  S _  \omega   ^ {c} $
with the variables  $  e ^ {x _ {i} } - 1 $, 
$  i = 1 \dots | \omega | $, 
and let  $  T \in H  ^ {**} ( {BU } ;  \mathbf Q ) $
be the characteristic class given by the product of the functions  $  x _ {i} / ( e ^ {x _ {i} } - 1 ) $. 
Let  $  x ( M) $
be the value of the characteristic class  $  x \in H  ^ {n} ( {BU } ;  \mathbf Q ) $
on the fundamental class  $  [ M] \in H _ {n} ( M , \mathbf Z ) $
of the almost-complex manifold  $  M $
with tangent bundle  $  T M $.

There exists for a homomorphism  $  \phi :  H  ^ {n} ( {BU } ;  \mathbf Q ) \rightarrow \mathbf Q $
a closed almost-complex manifold  $  M $
such that  $  \phi ( x) = x ( M) $
for all  $  x \in H  ^ {n} ( {BU } ;  \mathbf Q ) $
if and only if  $  \phi $
takes integer values on all the  $  n $-
dimensional components of each characteristic class  $  S _  \omega   ^ {c} ( e) T $(
Stong's theorem, see [[#References|[1]]], Chapt. 7). Equivalently, the Hurewicz homomorphism

$$ 
\pi _ {2 ( k + N ) } 
( T {BU } _ {N} )  \rightarrow \ 
\widetilde{K}  _ {2 ( k + N ) } 
( T {BU } _ {N} ) ,
$$

where  $  N \gg k $, 
is a monomorphism onto a direct summand (Hattori's theorem). Here  $  \widetilde{K}  $
denotes reduced [[K-theory| $  K $-
theory]].

2) Non-oriented, or orthogonal, cobordism. Each element of the ring  $  \Omega _ {O} $
has order  $  2 $, 
and

$$ 
\Omega _ {O}  =  \mathbf Z _ {2} [ x _ {1} \dots x _ {n} ,\dots ] ,\ \ 
 \mathop{\rm deg}  x _ {i}  =  - 1 ,\ \ 
i \neq 2  ^ {k} - 1 ,
$$

that is,  $  \Omega _ {O} $
is a free polynomial  $  \mathbf Z _ {2} $-
algebra. One can choose as generator  $  x _ {i} $
any element  $  [ M] $
with  $  S _ {(} i)  ^ {w} ( M) \neq 0 $, 
for example,  $  x _ {2i} = \mathbf R P  ^ {2i} $. 
In this theory there are analogues of the manifolds  $  H _ {i,j} $, 
obtained by replacing  $  \mathbf C P  ^ {k} $
by  $  \mathbf R P  ^ {k} $; 
a suitable manifold  $  H _ {i,j} $
can serve as a generator of degree  $  1 - i - j $. 
The Stiefel numbers (cf. [[Stiefel number|Stiefel number]]) completely define the non-orientable cobordism class of the manifold. The following theorem gives relations among the Stiefel numbers: Given a homomorphism  $  \phi :  H  ^ {n} ( {BO } , \mathbf Z _ {2} ) \rightarrow \mathbf Z _ {2} $, 
there exists a closed  $  n $-
dimensional manifold  $  M $
such that  $  \phi ( x) = x ( M) $
for all  $  x \in H  ^ {n} ( {BO } ;  \mathbf Z _ {2} ) $
if and only if  $  \phi ( S q b + v b ) = 0 $
for all  $  b \in H  ^ {*} ( {BO } ;  \mathbf Z _ {2} ) $, 
where  $  v = S q  ^ {-} 1 w $. 
Here  $  S q = S q  ^ {1} + S q  ^ {2} + \dots $
is the full [[Steenrod operation|Steenrod operation]] and  $  w = w _ {1} + w _ {2} + \dots $
is the full Stiefel class. The ring  $  ( \Omega _ {O} )  ^ {2} $
is the image of the homomorphism  $  \Omega _ {U} \rightarrow \Omega _ {O} $.

3) Oriented cobordism with ring  $  \Omega _ {SO} $. 
All the elements of the torsion subgroup  $   \mathop{\rm Tors} $
of this ring have order  $  2 $. 
The ring  $  \Omega _ {SO} /  \mathop{\rm Tors} $
is the ring of polynomials over  $  \mathbf Z $
of classes  $  x _ {i} $
of degree  $  - 4 i $, 
the generators being chosen by the condition

$$ 
S _ {(} i)  ^ {p} ( x _ {i} )  = \ 
\left \{ 
\begin{array}{lll}
\pm  1 ,  &2  ^ {i} \neq p  ^ {r} - 1  &\textrm{ for  any  "prime"  }  p  \textrm{ and  integer  }  r ,  \\
\pm  p ,  &2  ^ {i} = p  ^ {r} - 1  &\textrm{ for  some  "prime"  }  p  \textrm{ and  integer  }  r.  \\
\end{array}

 \right .$$

The  $  SO $-
cobordism class of a manifold is determined by the Pontryagin and Stiefel numbers (cf. [[Pontryagin number|Pontryagin number]]). The [[Signature|signature]] of the manifold is also an invariant of the cobordism class. The relations among the Stiefel numbers follow from the following fact: The image of the "forgetful" homomorphism  $  \Omega _ {SO} \rightarrow \Omega _ {O} $
consists precisely of those cobordism classes for which all numbers containing the class  $  w _ {1} $
are zero. For any partition  $  \omega = ( i _ {1} \dots i _ {k} ) $,

$$ 
p _  \omega  ( M)   \mathop{\rm mod}  2  = \ 
w _ {2 \omega }   ^ {2}  = \ 
[ w _ {2 i _ {1}  } \dots w _ {2 i _ {k}  } ( M) ] ,
$$

where  $  p _  \omega  $
is the corresponding Pontryagin number. There do not exist any  $  2 $-
prime relations among the Pontryagin numbers.

Similarly to the introduction of the classes  $  S _  \omega   ^ {c} ( e) $
for the unitary cobordism, the classes  $  S _  \omega   ^ {p} ( e) $
are introduced, which are symmetric functions in  $  e ^ {x _ {i} } + e ^ {- x _ {i} } - 2 $. 
Let  $  L $
be the characteristic class defining the Hirzebruch [[L-genus| $  L $-
genus]]. All relations among the Pontryagin numbers follow from the fact that the Pontryagin numbers are integers and  $  ( S _  \omega   ^ {p} ( e) L ) [ M] \in \mathbf Z [ 1/2 ] $. 
The homomorphism  $  \Omega _ {U} \rightarrow \Omega _ {SO} /  \mathop{\rm Tors} $
is epimorphic.

4) Special unitary cobordism with ring  $  \Omega _ {SU} $. 
A  $  U $-
manifold  $  M $
has an  $  SU $-
structure if and only if  $  c _ {1} ( M) = 0 $. 
All the elements of the torsion subgroup  $   \mathop{\rm Tors} $
have order 2. The kernel of the homomorphism  $  \Omega _ {SU} \rightarrow \Omega _ {U} $
is precisely  $   \mathop{\rm Tors} $. 
The group  $  \Omega _ {SU}  ^ {n} $
is finitely generated and  $  \Omega _ {SU} \otimes \mathbf Q $
is the ring of polynomials over  $  \mathbf Q $
of classes  $  x _ {i} $
of degree  $  - 2 i $, 
$  i > 1 $. 
The torsion subgroup  $   \mathop{\rm Tors} $
has the form  $   \mathop{\rm Tors}  ^ {-} n = 0 $
when  $  n \neq 8 k + 1 , 8 k + 2 $, 
while for  $  n = 8 k + 1 , 8 k + 2 $, 
$   \mathop{\rm Tors}  ^ {-} n $
is a vector space over  $  \mathbf Z _ {2} $
the dimension of which is the number of partitions of  $  k $. 
Two  $  SU $-
manifolds are bordant if and only if they have the same characteristic number in integer cohomology and in  $  KO $-
theory.

All relations among the Chern numbers for  $  n $-
dimensional  $  SU $-
manifolds follows from the following:  $  c _ {1} c _  \omega  ( M) = 0 $
for all  $  \omega $; 
$  ( S _  \omega   ^ {c} ( e) T) [ M] \in \mathbf Z $
for all  $  \omega $; 
if  $  n = 4 $
$   \mathop{\rm mod}  8 $, 
then  $  ( S _  \omega   ^ {p} ( e) T) [ M] \in 2 \mathbf Z $
for all  $  \omega $. 
The image of the homomorphism  $  \Omega _ {SU} \rightarrow \Omega _ {O} $
consists of the classes  $  [ M]  ^ {2} $, 
where  $  M $
is an oriented manifold all Pontryagin numbers of which containing the class  $  p _ {1} $
are even.

The rings  $  \Omega _ { \mathop{\rm Spin}  } $
and  $  \Omega _ { \mathop{\rm Spin}   \mathbf C } $
have also been completely computed. The rings  $  \Omega _ { \mathop{\rm Sp}  } $
and  $  \Omega _ { \mathop{\rm fr}  } $
have to date (1986) not been computed. The ring  $  \Omega _ { \mathop{\rm Sp}  } \otimes \mathbf Z [ 1 / 2 ] $
is the ring of polynomials on  $  ( - 4 i ) $-
dimensional generators. All known (1986) elements of  $   \mathop{\rm Tors}  \Omega _ { \mathop{\rm Sp}  } $
have order 2. (However there is an announcement of an element of order 4 in dimensions  $  > 100 $.) 
With regard to  $  \Omega _ { \mathop{\rm fr}  } $, 
the main result here is Serre's theorem on the finiteness of these groups. The ring  $  \Omega _ {S \mathbf C }  $
of self-adjoint cobordism has also been studied, where the objects are almost-complex manifolds with an operator given in the normal bundle which isomorphically maps the complex structure onto the adjoint. The spectrum of  $  TBS \mathbf C $
has been constructed; with regard to the groups  $  \Omega _ {SO} $
it is known that there is only  $  2 $-
prime torsion, but there are elements of order  $  4  ^ {k} $
for any  $  k $, 
namely  $  [ \mathbf R P  ^ {4k-} 3 ] $. 
The image  $   \mathop{\rm im} ( \Omega _ {S \mathbf C }  \rightarrow \Omega _ {O} ) $
has also been calculated using the technique of formal groups (cf. [[Formal group|Formal group]]).

A mapping of one cobordism theory into another, for example,  $  SU  ^ {*} \rightarrow U  ^ {*} $, 
induces a mapping of the spectra  $  TB SU \rightarrow T {BU } $. 
The cone of this mapping in the category of spectra gives a generalized cohomology theory. The ring of the point of the theory so obtained has the following geometric interpretation. Let  $  ( U , SU ) $
be a  $  U $-
manifold on the (possibly empty) boundary of which an  $  SU $-
structure is fixed. By introducing the appropriate bordism relation for  $  ( U , SU ) $-
manifolds one obtains the ring  $  \Omega _ {U , SU }  $. 
The groups  $  \Omega _ {U ,  \mathop{\rm fr}  } $, 
$  \Omega _ {O , SO }  $
etc., are introduced in the same way.

So far, smooth manifolds have been considered or, equivalently, linear group representations (the structure series arising from the bundles over  $  B SO _ {r} $). 
It is possible to consider various structures on topological manifolds, that is, to start from a group of homeomorphisms (and even proper homotopy equivalences) of  $  \mathbf R  ^ {r} $. 
Here the following examples are known. (Throughout, the letter  $  S $
denotes passage to the oriented case.)

5) Piecewise-linear cobordism. The objects are piecewise-linear manifolds. The corresponding bordism relation leads to the groups  $  \Omega _ { \mathop{\rm PL}  } $, 
$  \Omega _ {S  \mathop{\rm PL}  } $. 
By defining the group  $  { \mathop{\rm PL} } _ {n} $(
or  $  S { \mathop{\rm PL} } _ {n} $) 
as the group of piecewise-linear homeomorphisms of  $  \mathbf R  ^ {n} $
onto itself that preserve the origin (or the orientation as well), one can introduce the classifying spaces  $  B { \mathop{\rm PL} } _ {n} $(
or  $  BS { \mathop{\rm PL} } _ {n} $) 
and the Thom spaces  $  TB { \mathop{\rm PL} } _ {n} $(
or  $  TBS { \mathop{\rm PL} } _ {n} $) 
and construct a  $  { \mathop{\rm PL} } $(
or  $  S { \mathop{\rm PL} } $) 
cobordism theory. In this connection,  $  \Omega _ { \mathop{\rm PL}  }  ^ {-} i \approx \pi _ {i} ( TB { \mathop{\rm PL} }) $
and  $  \Omega _ {S} { \mathop{\rm PL} }  ^ {-} 1 \approx \pi _ {i} ( TBS { \mathop{\rm PL} } ) $. 
The groups  $  \Omega _ { \mathop{\rm PL}  } $
have been computed. The cobordism class of a piecewise-linear manifold is completely defined by the characteristic numbers, that is, by the elements of  $  H  ^ {*} ( B { \mathop{\rm PL} } ;  \mathbf Z _ {2} ) $.

6) Topological cobordism. The objects are topological manifolds for which the groups  $  \Omega _ { \mathop{\rm Top}  } $, 
$  \Omega _ {S  \mathop{\rm Top}  } $
are defined. By considering the group  $   \mathop{\rm Top} _ {n} $
of homeomorphisms of  $  \mathbf R  ^ {n} $
onto itself that preserve the origin, one can define the spaces  $  B  \mathop{\rm Top} $
and  $  TP  \mathop{\rm Top} _ {n} $. 
The groups  $  \pi _ {i} ( TP  \mathop{\rm Top} ) $
and  $  H  ^ {*} ( B \textrm{ Top  } , \mathbf Z _ {2} ) $
have been computed. However, the isomorphism  $  \Omega _ { \mathop{\rm Top}  }  ^ {-} i \approx \pi _ {i} ( TP  \mathop{\rm Top} ) $
has been established for all  $  i $
except  $  i = 4 $. 
The absence of a proof of this isomorphism is tied up with the fact that the transitivity theorem on which the isomorphism  $  \Omega _ {B, \phi }   ^ {-} i \approx \pi _ {i} ( T ( B , \phi )) $
is based for topological manifolds, has not been proved in the general case (but it has not been refuted either (1986)).

7) Cobordism of Poincaré complexes  $  \Omega _ {G} $, 
$  \Omega _ {SG} $. 
The objects are complexes with [[Poincaré duality|Poincaré duality]] and the bordism is the corresponding equivalence relation. Such complexes have a normal spherical bundle induced from the universal bundle over  $  B G _ {N} $(
or  $  B S G _ {N} $). 
Here  $  G _ {N} $(
or  $  S G _ {N} $) 
is an [[H-space| $  H $-
space]] of homotopy equivalences (of degree 1) of the sphere  $  S  ^ {N} $
onto itself. The Thom spectra  $  T B G $
and  $  T B S G $
to which these give rise have finite homotopy groups, whereas the signature defines a non-trivial homomorphism  $  \sigma :  \Omega _ {SG}  ^ {-} 4k \rightarrow \mathbf Z $, 
so that, a fortiori, the mapping  $  \Omega _ {SG}  ^ {-} i \rightarrow \pi _ {i+} N ( T B G _ {N} ) $
is not an isomorphism.

Yet another series of examples is given by cobordism of manifolds with singularities of a special type. (This is a very good technique for the construction of various cohomology theories with special properties.) One can construct along these lines a cobordism theory that is the same as ordinary singular cohomology theory and one that is the same as connected  $  K $-
theory.

The second stage in the development of cobordism theory is the study of cobordisms as specific generalized cohomology theories. Let  $  F $
denote one of the fields  $  \mathbf R , \mathbf C $
or the skew-field of quaternions  $  \mathbf H $, 
let  $  G F $
be the corresponding series of groups ( $  G \mathbf R _ {n} = O _ {n} $, 
$  G \mathbf C _ {n} = U _ {n} $, 
$  G \mathbf H _ {n} =  \mathop{\rm Sp} _ {n} $) 
and let  $  G F ^ { * } $
be the corresponding cobordism theory. A multiplicative generalized cohomology theory  $  h  ^ {*} $
is called  $  F $-
orientable if any  $  F $-
vector bundle is  $  h  ^ {*} $-
orientable or, equivalently, if the canonical one-dimensional  $  F $-
vector bundle  $  \xi \rightarrow F P  ^  \infty  $, 
where  $  F P  ^  \infty  $
is a projective space, is  $  h  ^ {*} $-
orientable. By an  $  F $-
orientation of the theory  $  h  ^ {*} $
one means an  $  h  ^ {*} $-
orientation  $  U _ {h} ( \xi ) \in h  ^ {*} ( F P  ^  \infty  ) $
of the bundle  $  \xi $, 
and a theory with a chosen orientation is called oriented. The  $  G F $-
cobordism theories have a canonical orientation because of the identification  $  F P  ^  \infty  = T B G F _ {1} $. 
The theory  $  G F  ^ {*} $
is universal in the class of  $  F $-
oriented theories, that is, for any  $  F $-
oriented theory  $  h  ^ {*} $
with  $  F $-
orientation  $  U _ {h} ( \xi ) $
there exists a unique multiplicative homomorphism of theories  $  \phi  ^ {h} :  G F ^ { * } \rightarrow h  ^ {*} $
under which the canonical orientation of the theory  $  G F ^ { * } $
is taken to  $  U _ {h} $. 
Moreover, when  $  F _ {O} $
is one of the fields  $  \mathbf R , \mathbf C $, 
there exist for any  $  F _ {O} $-
oriented theory  $  h  ^ {*} $
and any finite  $   \mathop{\rm CW} $-
complex  $  X $
spectral sequences  $  E _ {p,q}  ^ {r} ( X) $
and  $  E _ {r}  ^ {p,q} ( X) $
with

$$ 
E _ {p,q}  ^ {2} ( X)  = \ 
 \mathop{\rm Tor} _ {p,q} ^ {\Omega _ {G F _ {O}  } } ( G F _ {O} ^ { * }
( X) , h  ^ {*} (  \mathop{\rm pt} ) ) ,
$$

$$ 
E _ {2}  ^ {p,q} ( X)  =   \mathop{\rm Ext} _ {\Omega _ {G F _ {O}  } }  ^ {p,q} ( G F _ {O} ^ { * } ( X) , h  ^ {*} (  \mathop{\rm pt} ) ) ,
$$

converging to  $  h  ^ {*} ( X) $
and natural in  $  X $
and  $  h  ^ {*} $, 
where  $  h  ^ {*} (  \mathop{\rm pt} ) $
is made into an  $  \Omega _ {G F _ {O}  } $-
module by means of the homomorphism  $  \phi  ^ {h} (  \mathop{\rm pt} ) $. 
If  $  h _ {*} $
is the homology theory dual to the  $  F $-
oriented cohomology theory  $  h  ^ {*} $, 
then there is a homomorphism  $  \phi _ {h} :  G F _ {*} \rightarrow h _ {*} $. 
In the case when  $  h _ {*} $
is the ordinary homology theory, it coincides with Steenrod–Thom realization of cycles (see [[Steenrod problem|Steenrod problem]]). The powerful methods of cobordism theory are connected with formal groups (cf. [[Formal group|Formal group]], [[#References|[5]]]).

The most important and successful applications of cobordism theory are: the proof of the Atiyah–Singer index theorem for an elliptic operator and the general Riemann–Roch theorem; the study of fixed points of group actions; the classification of smooth (or piecewise-smooth) manifolds of given homotopy type; the proof of the theorem on the topological invariance of rational Pontryagin classes, and the solution of the problem of triangulability of topological manifolds.

See also the references in [[Bordism|Bordism]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> R.E. Stong, "Notes on cobordism theory" , Princeton Univ. Press (1968) {{MR|0248858}} {{ZBL|0181.26604}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> P.E. Conner, E.E. Floyd, "Differentiable periodic maps" , Springer (1964) {{MR|0176478}} {{ZBL|0125.40103}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> S.P. Novikov, "Methods of algebraic topology from the point of view of cobordism theory" ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''31''' : 4 (1967) pp. 855–951 (In Russian) {{MR|0221509}} {{ZBL|}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> T. Bröcker, T. Tom dieck, "Kobordismentheorie" , Springer (1970) {{MR|0275446}} {{ZBL|0211.55501}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> V.M. [V.M. Bukhshtaber] Buchstaber, "Cobordisms in problems of algebraic topology" ''J. Soviet Math.'' , '''7''' : 4 (1975) pp. 629–653 ''Itogi Nauk. i Tekh. Algebra. Geom. Topol.'' (1975) pp. 231–272</TD></TR></table>

====Comments====
The letter  $  M $
is often used to denote Thom spaces and Thom spectra. Thus, e.g.,  $  M U ( n) $
is used to denote the Thom space  $  T {BU } _ {n} $
and  $  M U $
stands for the spectrum of all the  $  M U ( n) $; 
similarly one uses  $  M S O ( n) $
for  $  TBSO _ {n} $, 
etc. The corresponding generalized cohomology theories are then indicated by the same symbols as is customary for generalized cohomology theories defined by a spectrum; thus,  $  M U  ^ {n} ( X) $
is the  $  n $-
th complex cobordism group of  $  X $
and  $  M U  ^ {*} ( X) $
is its complex cobordism ring.

A structural series  $  ( B , \phi ) $
as defined above is often called a  $  ( B , f  ) $-
structure (cf. [[B-Phi-structure| $  ( B , \phi ) $-
structure]], [[#References|[1]]]).

The general theorem that the (co)bordism group of  $  n $-
dimensional  $  ( B , f  ) $-
manifolds  $  \Omega _ {n} ( B , f  ) $
is isomorphic to  $  \lim\limits _ {r \rightarrow \infty }   \pi _ {n+} r ( T B _ {r} , \infty ) $
is known as the Pontryagin–Thom theorem.

A complex structure on a real vector bundle  $  E $
over a manifold  $  M $
is a vector bundle morphism  $  J :  E \rightarrow E $
such that  $  J  ^ {2} = - 1 $. 
If  $  M $
is a complex imbedded manifold without boundary  $  M \subset  \mathbf C  ^ {N} $, 
then multiplication with  $  i $
on its normal bundle defines a complex structure on that bundle (viewed as a real bundle). A weakly-complex manifold (also called a stably (almost) complex manifold) is a real manifold with a complex structure on its stable [[Normal bundle|normal bundle]]; i.e. if  $  F $
denotes the normal bundle of  $  M $, 
then there is a complex structure defined on some  $  F \oplus \theta  ^ {r} $
where  $  \theta  ^ {r} $
stands for the trivial  $  r $-
dimensional bundle over  $  M $, 
$  \theta  ^ {r} = M \times \mathbf R  ^ {r} $. 
The complex bordism groups of a space  $  S $, 
often denoted by  $  \underline{M U } {} _ {*} ( S) $, 
can now also be interpreted as cobordism classes of mappings  $  f :  M \rightarrow S $
where  $  M $
is a weakly-complex manifold without boundary. There is a similar interpretation of the complex cobordism groups  $  \underline{M U } {}  ^ {q} ( S) $, 
cf. [[#References|[a3]]], and for other bordism and cobordism group.

The relation between cobordism theory (and other (generalized) cohomology theories) and [[Formal group|formal group]] theory comes about as follows. A generalized cohomology theory  $  h $
is complex oriented if it has first Chern classes (in a suitable sense; cf. above and [[#References|[a1]]], p. 121; [[#References|[a5]]], Part II, (2.1)). Let  $  \xi $
be the class of the canonical line bundle  $  H $
over  $  \mathbf C P  ^  \infty  $, 
the space of lines in  $  \mathbf C  ^  \infty  $(
the fibre of  $  H $
at  $  x \in \mathbf C P  ^  \infty  $
is the line  $  x $). 
Then  $  h  ^ {*} ( \mathbf C P  ^  \infty  ) = h  ^ {*} (  \mathop{\rm pt} ) [ [ \xi ] ] $
and  $  h  ^ {*} ( \mathbf C P  ^  \infty  \times \mathbf C P  ^  \infty  ) = h  ^ {*} (  \mathop{\rm pt} ) [ [ \xi \otimes 1 , 1 \otimes \xi ] ] $. 
Because  $  \mathbf C P  ^  \infty  = \underline{BU } ( 1) $
is classifying for complex line bundles, there is an  $  m :  \mathbf C P  ^  \infty  \times \mathbf C P  ^  \infty  \rightarrow \mathbf C P  ^  \infty  $
such that  $  m  ^ {!} ( H) = H \otimes H $
and this induces a ring homomorphism  $  h  ^ {*} ( \mathbf C P  ^  \infty  ) \rightarrow h  ^ {*} ( \mathbf C P  ^  \infty  \times \mathbf C P  ^  \infty  ) $. 
The image of  $  \xi $
is a power series in two variables, here denoted by  $  F _ {h} ( X , Y ) = \sum _ {i,j} a _ {ij} X  ^ {i} Y  ^ {j} $, 
$  a _ {ij} \in h  ^ {*} (  \mathop{\rm pt} ) $, 
where  $  X $
stands for  $  \xi \otimes 1 $
and  $  Y $
for  $  1 \otimes \xi $. 
Equivalently,  $  F _ {h} $
is the power series with coefficients in  $  h (  \mathop{\rm pt} ) $
such that  $  c _ {1} ( H \otimes H ) = F _ {h} ( c _ {1}  ^ {h} ( H \otimes 1 ) , c _ {1}  ^ {h} ( 1 \otimes H ) ) $. 
The power series  $  F _ {h} $
defines a [[Formal group|formal group]] law.

Conversely, the question arises whether every (one-dimensional commutative) formal group arises as the formal group of a generalized cohomology theory. Here the study of (co)bordism of manifolds with special singularities is important.

It turns out (D. Quillen [[#References|[a4]]], cf. also [[#References|[a12]]]) that for  $  h = \underline{M U } $
this formal group law  $  F _ {h} $
is a universal formal group law. This universality property shows up in topological terms in the form of the theorem that if  $  h  ^ {*} $
is any complex oriented generalized cohomology theory, then there is a unique transformation of cohomology theories  $  \phi :  \underline{M U } {}  ^ {*} \rightarrow h  ^ {*} $(
linear, degree preserving and multiplicative) such that  $  \phi _ {*} F _ {\underline{M U }  } ( X , Y ) = F _ {h} ( X , Y ) $, 
where  $  \phi _ {*} $
means: apply  $  \phi (  \mathop{\rm pt} ) $
to the coefficients of  $  F _ {\underline{M U }  } ( X , Y ) $. 
The logarithm of the formal group law  $  F _ {\underline{M U }  } $
can be calculated (A.S. Mishchenko, cf. [[#References|[a12]]]; cf. [[Formal group|Formal group]] for the notion of logarithm of a formal group law). It is equal to

$$ 
 \mathop{\rm log}  F _ {\underline{M U }  } ( X)  = \ 
\sum _ { i= } 1 ^  \infty  
m _ {i} X  ^ {i} ,
$$

$$ 
m _ {1} = 1 ,\  m _ {i} = i  ^ {-} 1 [ \mathbf C P  ^ {i-} 1 ] ,\  i \geq  2 ,
$$

where  $  [ \mathbf C P  ^ {n} ] \in \underline{M U } {}  ^ {*} (  \mathop{\rm pt} ) $
denotes the complex cobordism class of the complex projective space of (complex) dimension  $  n $.

On the other hand it is possible to write down explicit formulas for the logarithm of a universal formal group law over  $  \mathbf Z [ u _ {2} , u _ {3} ,\dots ] $, 
cf. [[#References|[a2]]], Chapt. 1. There result explicit formulas for the polynomial generators of  $  \underline{M U } {}  ^ {*} (  \mathop{\rm pt} ) = \mathbf Z [ u _ {1} , u _ {2} ,\dots ] $
in terms of the  $  [ \mathbf C P  ^ {n} ] $. 
These formulas take a particularly useful form for the "p-typical" version  $  \underline{B P } {}  ^ {*} $
of the cohomology theory  $  \underline{M U } {}  ^ {*} $. 
The generalized cohomology theory  $  \underline{B P } {}  ^ {*} $, 
Brown–Peterson cohomology for a prime number  $  p $, 
cf. [[#References|[a6]]], is defined by a spectrum  $  \underline{B P } $
and is such that  $  \underline{B P } (  \mathop{\rm pt} ) = \mathbf Z _ {(} p) [ v _ {1} , v _ {2} ,\dots ] $
with  $  v _ {i} $
of degree  $  - 2 ( p  ^ {i} - 1 ) $. 
It is such that  $  \underline{M U } {}  ^ {*} ( X) \oplus \mathbf Z _ {(} p) $
is a direct sum of (dimension shifted) copies of  $  \underline{B P } {}  ^ {*} ( X) $
for each space  $  X $, 
functorially in  $  X $. 
Here  $  \mathbf Z _ {(} p) $
stands for the ring of integers localized at  $  p $, 
i.e.  $  \mathbf Z _ {(} p) = \{ {a/b \in Q } : {( p , b ) = 1 } \} $. 
The theory  $  \underline{B P } {}  ^ {*} $
can also be defined as the image of an idempotent cohomology operator  $  \underline{M U } {}  ^ {*} \oplus \mathbf Z _ {(} p) \rightarrow \underline{M U } {}  ^ {*} \oplus \mathbf Z _ {p} $(
cf., e.g., [[#References|[a1]]], Chapt. 4). This operation corresponds to  $  p $-
typification in formal group theory. The Hazewinkel generators ([[#References|[a1]]], pp. 137, 369-370)  $  v _ {1} , v _ {2} ,\dots $
of  $  \underline{B P }  ^ {*} (  \mathop{\rm pt} ) $
are defined recursively by

$$ 
p m _ {p  ^ {n}  - 1 } =
$$

$$ 
= \ 
v _ {n} + m _ {p-} 1 v _ {n-} 1  ^ {p} + m _ {p  ^ {2}  - 1 } v _ {n-} 2 ^ {
p  ^ {2} } + \dots + m _ {p  ^ {n-}  1 - 1 } v _ {1} ^ {p  ^ {n-} 1 } .
$$

They arise from the explicit construction of a  $  p $-
typical universal formal group [[#References|[a8]]]. A different set of generators  $  \overline{v}\; _ {1} \dots \overline{v}\; _ {i} \equiv v _ {i} $
$   \mathop{\rm mod}  p $
has been given by S. Araki [[#References|[a7]]], the Araki generators.

In a certain precise sense,  $  \underline{B P } $-
theory is  $  \underline{M U } $-
theory for one prime at the time, and currently a great deal of complex cobordism theory is written in terms of  $  \underline{B P } $
rather than  $  \underline{M U } $
itself. Combined with the theory of cohomology operations, formal group theory (the rings of operations  $  \underline{M U } {}  ^ {*} ( \underline{M U } ) $
and  $  \underline{B P } {}  ^ {*} ( \underline{B P } ) $
of  $  \underline{M U } $
and  $  \underline{B P } $
also have interpretations in terms of formal groups, cf. [[#References|[a1]]], [[#References|[a9]]], [[#References|[a10]]]), and spectral sequences, notably the Adams–Novikov spectral sequence and the chromatic spectral sequence (cf. [[#References|[a1]]], [[#References|[a11]]]), complex cobordism and Brown–Peterson cohomology have become a strong calculation tool in algebraic topology, e.g. for the stable homotopy groups of the spheres.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> D.C. Ravenel, "Complex cobordism and stable homotopy groups of spheres" , Acad. Press (1986) {{MR|0860042}} {{ZBL|0608.55001}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> M. Hazewinkel, "Formal groups and applications" , Acad. Press (1978) {{MR|0506881}} {{MR|0463184}} {{ZBL|0454.14020}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> D. Quillen, "Elementary proofs of some results of cobordism theory using Steenrod operations" ''Adv. Math.'' , '''7''' (1971) pp. 29–56 {{MR|0290382}} {{ZBL|0214.50502}} </TD></TR><TR><TD valign="top">[a4]</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 {{MR|0253350}} {{ZBL|0199.26705}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> J.F. Adams, "Stable homotopy and generalised homology" , Univ. Chicago Press (1974) pp. Part III, Chapt. 12 {{MR|0402720}} {{ZBL|0309.55016}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> E.H. Brown, F.P. Peterson, "A spectrum whose $\ZZ_p$-cohomology is the algebra of reduced $p$-th powers" ''Topology'' , '''5''' (1966) pp. 149–154</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> S. Araki, "Typical formal groups in complex cobordism and K-theory" , Kinokuniya (1973) {{MR|0375354}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> M. Hazewinkel, "Constructing formal groups III: applications to complex cobordism and Brown–Peterson cohomology" ''J. Pure Appl. Algebra'' , '''10''' (1977) pp. 1–18 {{MR|0463182}} {{MR|0463183}} {{MR|0463184}} {{ZBL|0363.14013}} </TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> P.S. Landweber, "$BP^\ast(BP)$ and typical formal groups" ''Osaka J. Math.'' , '''12''' (1975) pp. 357–363 {{MR|377945}} {{ZBL|0311.55003}} </TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top"> P.S. Landweber, "Invariant regular ideals in Brown–Peterson cohomology" ''Duke Math. J.'' , '''42''' (1975) pp. 499–505</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top"> H.R. Miller, D.C. Ravenel, W.S. Wilson, "Periodic phenomena of the Adams–Novikov spectral sequence" ''Ann. of Math.'' , '''106''' (1977) pp. 469–516 {{MR|458423}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top"> V.M. Bukhshtaber, A.S. Mishchenko, S.P. Novikov, "Formal groups and their role in the apparatus of algebraic topology" ''Russ. Math. Surveys'' , '''26''' (1971) pp. 63–90 ''Uspekhi Mat. Nauk'' , '''26''' (1971) pp. 131–154 {{MR|}} {{ZBL|0224.57006}} </TD></TR></table>