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A [[Characteristic number|characteristic number]] of a quasi-complex manifold. Let  $  x \in H  ^ {**} (  \mathop{\rm BU} _ {n} ) $
be an arbitrary characteristic class. For a closed quasi-complex manifold  $  M  ^ {2n} $
the integer  $  x [ M  ^ {2n} ] = \langle  x ( \tau M ) , [ M  ^ {2n} ] \rangle $
is called the Chern number of the manifold  $  M  ^ {2n} $
corresponding to the class  $  x $. 
Here  $  [ M  ^ {2n} ] \in H _ {2n} ( M  ^ {2n} ) $
is the [[Fundamental class|fundamental class]] of the manifold, or the orientation, uniquely determined by the quasi-complex structure, and  $  \tau M $
is the tangent bundle of  $  M $. 
If  $  x $
is taken to be a characteristic class with rational coefficients, then the corresponding Chern number will be rational. The Chern number  $  x [ M  ^ {2n} ] $
depends only on the homogeneous component of  $  x $
of degree  $  2n $. 
The Chern numbers are quasi-complex bordism invariants, and hence the characteristic class  $  x $
induces a homomorphism  $  \Omega _ {2n}  ^ {u} \rightarrow \mathbf Z $.

A partition of a number  $  n $
is a set  $  \omega = \{ i _ {1} \dots i _ {k} \} $
of non-negative integers with  $  i _ {1} + \dots + i _ {k} = n $. 
If  $  M , N $
are two quasi-complex manifolds of dimension  $  2n $
such that  $  c _  \omega  [ M] = c _  \omega  [ N ] $(
cf. [[Chern class|Chern class]]) for all partitions  $  \omega $
of  $  n $, 
then the manifolds  $  M , N $
are cobordant (in the quasi-complex sense).

Let  $  A $
be a free Abelian group with basis  $  \{ e _  \omega  \} = \{ e _ {i _ {1}  \dots i _ {k} } \} $
in one-to-one correspondence with the set of all partitions of  $  n $. 
The cited theorem asserts that the homomorphism

$$ 
\phi :  \Omega _ {2n}  ^ {u}  \rightarrow  A ,\ \ 
\phi ( [ M  ^ {2n} ] )  =  \sum _  \omega  c _  \omega  [ M  ^ {2n} ] e _  \omega   $$

is a monomorphism. Below a description of the image of the homomorphism  $  \phi $
is given (the Milnor–Hirzebruch problem). In other words, which sets of integers  $  a _  \omega  = a _ {i _ {1}  \dots i _ {k} } $, 
defined for all partitions  $  \omega $
of a number  $  n $, 
arise as the Chern numbers of quasi-complex manifolds? A Chern number can be defined in an arbitrary multiplicative oriented cohomology theory  $  h  ^ {*} $, 
except that in this case the Chern number of a quasi-complex manifold will be an element of the ring  $  h  ^ {*} (  \mathop{\rm pt} ) $. 
Dual to the cohomology theory  $  h  ^ {*} $
is a homology theory  $  h _ {*} $, 
and since  $  h  ^ {*} $
is oriented and multiplicative, there is for each quasi-complex manifold  $  M $
a unique fundamental class  $  [ M , \partial  M ]  ^ {h} \in h _ {2n} ( M , \partial  M ) $, 
where  $  2n =  \mathop{\rm dim}  M $. 
Moreover, as in the ordinary theory there is a pairing

$$ 
h  ^ {n} ( M , \partial  M ) \otimes h _ {m} ( M ,\ 
\partial  M )  \rightarrow  h  ^ {n-m} (  \mathop{\rm pt} ) .
$$

If  $  x \in H  ^ {*} ( M , \partial  M ) $, 
then the application of  $  x $
to  $  [ M , \partial  M ]  ^ {h} $
with respect to this pairing is denoted by  $  \{ x , [ M , \partial  M ]  ^ {h} \} \in h  ^ {*} (  \mathop{\rm pt} ) $. 
For a characteristic class  $  y $
with values in  $  h  ^ {*} $
and a closed quasi-complex manifold  $  M $, 
the element  $  \{ y ( \tau M ) , [ M ]  ^ {h} \} $
is called the Chern number in the theory  $  h  ^ {*} $. 
The preceding considerations apply also to [[K-theory| $  K $-
theory]]. Let  $  M $
be a quasi-complex manifold (possibly with boundary), let  $   \mathop{\rm dim} _ {\mathbf R }   M = 2n $
and let  $  x $
be an arbitrary element of  $  K  ^ {0} ( M , \partial  M ) $. 
Then the integer

$$ 
\{ x , [ M , \partial  M ]  ^ {k} \}  \in  K  ^ {-2n}
(  \mathop{\rm pt} )  \cong ^ { {\beta  ^ {n}} }  K  ^ {0} (  \mathop{\rm pt} )  =  \mathbf Z
$$

can be computed according to the formula

$$ 
\{ x , [ M , \partial  M ]  ^ {k} \}  = \ 
\langle   \mathop{\rm ch}  x T ( \tau M ) , [ M , \partial  M ] \rangle ,
$$

where  $  T $
is the [[Todd class]] given by the series  $\prod_{i=1}^ {n} x _ {i} / ( 1 - e ^ {x _ {i} } ) $. 
If the manifold  $  M $
is closed, then putting  $  x = 1 \in K  ^ {0} ( M) $
one obtains  $  \{ 1 , [ M ]  ^ {k} \} = T [ M] $. 
The characteristic number  $  T [ M ] $
is called the Todd genus of the manifold  $  M $
and is an integer for any quasi-complex manifold  $  M $. 
$  T [ M ] $
is often denoted by  $   \mathop{\rm Td} ( M) $.

One of the most important examples of a quasi-complex manifold is a tangent manifold. Let  $  N $
be a closed real manifold of dimension  $  n $. 
The manifold  $  TN $
of all tangent vectors to  $  N $
has a natural quasi-complex structure:  $  \tau T N = \tau N \oplus N $, 
$  i ( x , y ) = ( y , - x ) $. 
Fix a Riemannian metric on  $  N $
and let  $  BN \subset  TN $
denote the manifold with boundary consisting of all vectors of length not exceeding one. If  $  \sigma \in K  ^ {0} ( BN , \partial  BN ) $, 
then the integer  $  i _ {t} ( \sigma ) = \{ \sigma , [ BN , \partial  BN ]  ^ {k} \} $
is called the topological index of the element  $  \sigma $. 
If  $  \sigma $
is the class of the symbol of an elliptic operator  $  D $
defined on  $  N $, 
then  $  \textrm{ index }  D = i _ {t} ( \sigma ) $(
the Atiyah–Singer theorem), and applying the above formula for computing the integer  $  \{ x , [ M , \partial  M ]  ^ {k} \} $
leads to the cohomological form of the index theorem.

For a set  $  \omega = \{ i _ {1} \dots i _ {n} \} $
of non-negative integers and a closed quasi-complex manifold  $  M $
of dimension  $  2n $, 
let  $  S _  \omega   ^ {k} [ M] $
be the Chern number in  $  K $-
theory:

$$ 
S _  \omega   ^ {k} [ M]  =  S _  \omega  ( \gamma _ {1} \dots
\gamma _ {n} ) [ M] =
$$

$$ 
= \ 
\{ S _  \omega  ( \gamma _ {1} \dots \gamma _ {n} ) ( \tau M ) , [ M]  ^ {k} \} ,
$$

and let  $  S _  \omega  [ M] $
be the ordinary Chern number  $  S _  \omega  ( c _ {1} \dots c _ {n} ) [ M] $. 
The number  $  S _  \omega  [ M] $
can be distinct from zero only if  $  \omega $
is a partition of  $  n $. 
The number  $  S _  \omega   ^ {k} [ M] $
can be distinct from zero for sets  $  \omega = \{ i _ {1} \dots i _ {k} \} $
with  $  i _ {1} + \dots + i _ {k} \leq  n $. 
Any homomorphisms  $  \Omega _ {2n}  ^ {u} \rightarrow \mathbf Z $
can be expressed as a linear combination with integer coefficients of homomorphisms  $  S _  \omega   ^ {k} :  \Omega _ {2n}  ^ {u} \rightarrow \mathbf Z $, 
with  $  | \omega | \leq  n $, 
where  $  | \omega | = i _ {1} + \dots i _ {k} $(
the Stong–Hattori theorem). The characteristic numbers  $  S _  \omega   ^ {k} [ M] $
with  $  | \omega | \leq  n $
can be expressed in the form

$$ 
S _  \omega   ^ {k} [ M]  =  \sum _
{| \omega  ^  \prime  | = n } r _ {\omega  ^  \prime   } c _ {\omega  ^  \prime   } [ M] ,
$$

where  $  r _ {\omega  ^  \prime   } $
are rational coefficients and  $  M $
is any closed quasi-complex manifold of dimension  $  2n $. 
Let  $  a $
be an arbitrary element of the group  $  A $, 
$  a= \sum _ {| \omega  ^  \prime   | = n } a _ {\omega  ^  \prime   } e _ {\omega  ^  \prime   } $
and let  $  S _  \omega   ^ {k} ( a) = \sum _ {| \omega  ^  \prime   | = n } r _ {\omega  ^  \prime   } a _ {\omega  ^  \prime   } $. 
Then the element  $  a \in A $
lies in the image of the homomorphism  $  \phi :  \Omega _ {2n}  ^ {u} \rightarrow A $
if and only if  $  S _  \omega   ^ {k} $
is an integer for all sets  $  \omega $
with  $  | \omega | \leq  n $.

For references see [[Chern class|Chern class]].

====Comments====
Cf. [[Cobordism|Cobordism]] for the notions  "quasi-complex manifold"  and  "complex-oriented cohomology theory" . Cf. also the comments to [[Chern class|Chern class]].