# Chern number

A 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 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) 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. 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.