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
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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
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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
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can be computed according to the formula
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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:
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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
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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.
Chern number. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Chern_number&oldid=46334