Chern number

A characteristic number of a quasi-complex manifold. Let be an arbitrary characteristic class. For a closed quasi-complex manifold the integer is called the Chern number of the manifold corresponding to the class . Here is the fundamental class of the manifold, or the orientation, uniquely determined by the quasi-complex structure, and is the tangent bundle of . If is taken to be a characteristic class with rational coefficients, then the corresponding Chern number will be rational. The Chern number depends only on the homogeneous component of of degree . The Chern numbers are quasi-complex bordism invariants, and hence the characteristic class induces a homomorphism .

A partition of a number is a set of non-negative integers with . If are two quasi-complex manifolds of dimension such that (cf. Chern class) for all partitions of , then the manifolds are cobordant (in the quasi-complex sense).

Let be a free Abelian group with basis in one-to-one correspondence with the set of all partitions of . The cited theorem asserts that the homomorphism

is a monomorphism. Below a description of the image of the homomorphism is given (the Milnor–Hirzebruch problem). In other words, which sets of integers , defined for all partitions of a number , arise as the Chern numbers of quasi-complex manifolds? A Chern number can be defined in an arbitrary multiplicative oriented cohomology theory , except that in this case the Chern number of a quasi-complex manifold will be an element of the ring . Dual to the cohomology theory is a homology theory , and since is oriented and multiplicative, there is for each quasi-complex manifold a unique fundamental class , where . Moreover, as in the ordinary theory there is a pairing

If , then the application of to with respect to this pairing is denoted by . For a characteristic class with values in and a closed quasi-complex manifold , the element is called the Chern number in the theory . The preceding considerations apply also to -theory. Let be a quasi-complex manifold (possibly with boundary), let and let be an arbitrary element of . Then the integer

can be computed according to the formula

where is the Todd class given by the series . If the manifold is closed, then putting one obtains . The characteristic number is called the Todd genus of the manifold and is an integer for any quasi-complex manifold . is often denoted by .

One of the most important examples of a quasi-complex manifold is a tangent manifold. Let be a closed real manifold of dimension . The manifold of all tangent vectors to has a natural quasi-complex structure: , . Fix a Riemannian metric on and let denote the manifold with boundary consisting of all vectors of length not exceeding one. If , then the integer is called the topological index of the element . If is the class of the symbol of an elliptic operator defined on , then (the Atiyah–Singer theorem), and applying the above formula for computing the integer leads to the cohomological form of the index theorem.

For a set of non-negative integers and a closed quasi-complex manifold of dimension , let be the Chern number in -theory:

and let be the ordinary Chern number . The number can be distinct from zero only if is a partition of . The number can be distinct from zero for sets with . Any homomorphisms can be expressed as a linear combination with integer coefficients of homomorphisms , with , where (the Stong–Hattori theorem). The characteristic numbers with can be expressed in the form

where are rational coefficients and is any closed quasi-complex manifold of dimension . Let be an arbitrary element of the group , and let . Then the element lies in the image of the homomorphism if and only if is an integer for all sets with .

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.