Difference between revisions of "Chern class"
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====References==== | ====References==== | ||
− | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> | + | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> S.S. Chern, "Characteristic classes of Hermitian manifolds" ''Ann. of Math.'' , '''47''' : 1 (1946) pp. 85–121 {{MR|0015793}} {{ZBL|0060.41416}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> R.E. Stong, "Notes on cobordism theory" , Princeton Univ. Press (1968) {{MR|0248858}} {{ZBL|0181.26604}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> R.S. Palais, "Seminar on the Atiyah–Singer index theorem" , Princeton Univ. Press (1965) {{MR|0198494}} {{ZBL|0137.17002}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> P.E. Conner, E.E. Floyd, "Differentiable periodic maps" , Springer (1964) {{MR|0176478}} {{ZBL|0125.40103}} </TD></TR><TR><TD valign="top">[5a]</TD> <TD valign="top"> M.F. Atiyah, I.M. Singer, "The index of elliptic operators I" ''Ann. of Math. (2)'' , '''87''' (1968) pp. 484–530 {{MR|0236950}} {{MR|0232402}} {{ZBL|0164.24001}} </TD></TR><TR><TD valign="top">[5b]</TD> <TD valign="top"> M.F. Atiyah, G.B. Segal, "The index of elliptic operators II" ''Ann. of Math. (2)'' , '''87''' (1968) pp. 531–545 {{MR|0236953}} {{MR|0236951}} {{ZBL|0164.24201}} </TD></TR><TR><TD valign="top">[5c]</TD> <TD valign="top"> M.F. Atiyah, I.M. Singer, "The index of elliptic operators III" ''Ann. of Math. (2)'' , '''87''' (1968) pp. 546–604 {{MR|0236952}} {{ZBL|0164.24301}} </TD></TR><TR><TD valign="top">[5d]</TD> <TD valign="top"> M.F. Atiyah, I.M. Singer, "The index of elliptic operators IV" ''Ann. of Math. (2)'' , '''93''' (1971) pp. 119–138 {{MR|0279833}} {{ZBL|0212.28603}} </TD></TR><TR><TD valign="top">[5e]</TD> <TD valign="top"> M.F. Atiyah, I.M. Singer, "The index of elliptic operators V" ''Ann. of Math. (2)'' , '''93''' (1971) pp. 139–149 {{MR|0279834}} {{ZBL|0212.28603}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> F. Hirzebruch, "Topological methods in algebraic geometry" , Springer (1978) (Translated from German) {{MR|1335917}} {{MR|0202713}} {{ZBL|0376.14001}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> D. Husemoller, "Fibre bundles" , McGraw-Hill (1966) {{MR|0229247}} {{ZBL|0144.44804}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> V.M. Bukhshtaber, "The Chern–Dold character in cobordisms" ''Math. USSR-Sb.'' , '''12''' : 4 (1970) pp. 573–594 ''Mat. Sb.'' , '''83''' (1970) pp. 575–595 {{MR|}} {{ZBL|0219.57027}} </TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> S.P. Novikov, "The method of algebraic topology from the viewpoint of cobordism theory" ''Math. USSR-Izv.'' , '''4''' : 1 (1967) pp. 827–913 ''Izv. Akad. SSSR Ser. Mat.'' , '''31''' : 4 (1967) pp. 855–951</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top"> M.F. Atiyah, "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c022030127.png" />-theory: lectures" , Benjamin (1967) {{MR|224083}} {{ZBL|}} </TD></TR></table> |
Revision as of 21:50, 30 March 2012
A characteristic class defined for complex vector bundles. A Chern class of the complex vector bundle over a base
is denoted by
and is defined for all natural indices
. By the complete Chern class is meant the inhomogeneous characteristic class
, and the Chern polynomial is the expression
, where
is a formal unknown. Chern classes were introduced in [1].
The characteristic classes, defined for all -dimensional complex vector bundles and with values in the integral cohomology, are naturally identified with the elements of the ring
. In this sense the Chern classes
can be thought of as elements of the groups
, the complete Chern class as an element of the ring
, and the Chern polynomial as an element of the formal power series ring
.
The Chern classes satisfy the following properties, which uniquely determine them. 1) For two vector bundles with a common base
,
, in other words
where
. 2) For the one-dimensional universal bundle
over
the identity
holds, where
is the orientation of
(
is the Thom space of
, which, being complex, has a uniquely-defined orientation
).
Consequences of the properties 1)–2) are: for
, and
, where
is the trivial bundle. The latter fact allows one to define Chern classes as elements of the ring
.
If is a collection of non-negative integers, then
denotes the characteristic class
, where
.
Under the natural monomorphism induced by the mapping
, the Chern classes are mapped into the elementary symmetric functions, and the complete Chern class is mapped to the polynomial
. The image of the ring
in
is the subring consisting of all symmetric formal power series. Every symmetric formal power series in the Wu generators
determines a characteristic class that can be expressed in terms of Chern classes. For example, the series
determines a characteristic class with rational coefficients, called the Todd class and denoted by
.
Let be a set of non-negative integers. Let
denote the characteristic class defined by the smallest symmetric polynomial in the variables
, where
, containing the monomial
.
Let be an oriented multiplicative cohomology theory. Then the Chern classes
with values in
satisfy, as do ordinary Chern classes, the properties:
,
,
, where
is the orientation of the bundle
, and these properties completely determine them. As with ordinary Chern classes, one uses the notation
and
. If
are two complex vector bundles, then
![]() |
![]() |
where the summation is taken over all sets with
.
In place of the theory one may take a unitary cobordism theory
or
-theory. For a
-theory the element
is defined by the identity mapping
, and for
-theory
, where
is the Bott periodicity operator. The notation
is retained for Chern classes with values in a
-theory, while Chern classes with values in
-theory are denoted by
.
According to the general theory, , where
is a vector bundle with base
. However
-theory is often conveniently thought of as a
-graded theory, identifying the groups
and
via the periodicity operator
. Then
and
for all
. From this point of view it makes sense to consider, instead of the complete Chern class, the Chern polynomial
![]() |
Let be a cohomology operation in
-theory (
terms). The polynomial
![]() |
satisfies, as does , the multiplicative property
![]() |
There is the following connection between these polynomials:
![]() |
Here both parts of the equation lie in and
is the trivial bundle of dimension
. The classes
in this construction are different from those constructed by M.F. Atiyah, who defined them by the formula
. R. Stong [2] defined classes
that satisfy the condition
![]() |
The difference arises because, for Stong,
![]() |
The classes are connected with the notion of a Landweber–Novikov algebra, which is very fruitful in homotopy theory. For an arbitrary set
of non-negative integers, consider the characteristic class
, where
. There is a Thom isomorphism
, where
is the spectrum corresponding to the
-theory. The image of the class
in
determines a cohomology operation in the
-theory. The subalgebra of the Steenrod algebra in the
-theory generated by the operations of this form is called the Landweber–Novikov algebra. The operation constructed from the set
is denoted by
.
For one-dimensional bundles there is the identity
![]() |
This important property, which enables one to define the Chern character, does not hold in generalized cohomology theories. However there exists a formal power series with coefficients in
, such that
, where
is the first Chern class with coefficients in
. For the unitary cobordism theory
![]() |
where is the cobordism class of the projective space
. This series is called the Mishchenko series.
References
[1] | S.S. Chern, "Characteristic classes of Hermitian manifolds" Ann. of Math. , 47 : 1 (1946) pp. 85–121 MR0015793 Zbl 0060.41416 |
[2] | R.E. Stong, "Notes on cobordism theory" , Princeton Univ. Press (1968) MR0248858 Zbl 0181.26604 |
[3] | R.S. Palais, "Seminar on the Atiyah–Singer index theorem" , Princeton Univ. Press (1965) MR0198494 Zbl 0137.17002 |
[4] | P.E. Conner, E.E. Floyd, "Differentiable periodic maps" , Springer (1964) MR0176478 Zbl 0125.40103 |
[5a] | M.F. Atiyah, I.M. Singer, "The index of elliptic operators I" Ann. of Math. (2) , 87 (1968) pp. 484–530 MR0236950 MR0232402 Zbl 0164.24001 |
[5b] | M.F. Atiyah, G.B. Segal, "The index of elliptic operators II" Ann. of Math. (2) , 87 (1968) pp. 531–545 MR0236953 MR0236951 Zbl 0164.24201 |
[5c] | M.F. Atiyah, I.M. Singer, "The index of elliptic operators III" Ann. of Math. (2) , 87 (1968) pp. 546–604 MR0236952 Zbl 0164.24301 |
[5d] | M.F. Atiyah, I.M. Singer, "The index of elliptic operators IV" Ann. of Math. (2) , 93 (1971) pp. 119–138 MR0279833 Zbl 0212.28603 |
[5e] | M.F. Atiyah, I.M. Singer, "The index of elliptic operators V" Ann. of Math. (2) , 93 (1971) pp. 139–149 MR0279834 Zbl 0212.28603 |
[6] | F. Hirzebruch, "Topological methods in algebraic geometry" , Springer (1978) (Translated from German) MR1335917 MR0202713 Zbl 0376.14001 |
[7] | D. Husemoller, "Fibre bundles" , McGraw-Hill (1966) MR0229247 Zbl 0144.44804 |
[8] | V.M. Bukhshtaber, "The Chern–Dold character in cobordisms" Math. USSR-Sb. , 12 : 4 (1970) pp. 573–594 Mat. Sb. , 83 (1970) pp. 575–595 Zbl 0219.57027 |
[9] | S.P. Novikov, "The method of algebraic topology from the viewpoint of cobordism theory" Math. USSR-Izv. , 4 : 1 (1967) pp. 827–913 Izv. Akad. SSSR Ser. Mat. , 31 : 4 (1967) pp. 855–951 |
[10] | M.F. Atiyah, "![]() |
Comments
denotes the completion
of
.
The power series for a complex oriented cohomology theory
such that
is the logarithm of the formal group
defined by
; cf. Cobordism and Formal group for some more details.
Chern class. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Chern_class&oldid=15158