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.

Comments

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