In mathematics, in particular in algebraic topology and differential geometry, the Chern classes (pronounced chen) are a particular type of characteristic class associated to complex vector bundles. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics on Wiktionary, the free dictionary Wikimedia Commons has media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. ... In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ... In mathematics, the idea of characteristic class is one of the unifying geometric concepts in algebraic topology, differential geometry and algebraic geometry. ... In mathematics, a vector bundle is a geometrical construct where to every point of a topological space (or manifold, or algebraic variety) we attach a vector space in a compatible way, so that all those vector spaces, glued together, form another topological space (or manifold or variety). ...

Chern classes are named for Shiing-Shen Chern, who first gave a general definition of them in the 1940s. Chen Xingshen Shiing-shen Chern (陳省身; pinyin: Chén Xǐngshēn; October 26, 1911–December 3, 2004) was a Chinese mathematician, who was one of the leading differential geometers of the twentieth century. ... Jump to: navigation, search // Events and trends The 1940s were seen as a transition period between the radical 1930s and the conservative 1950s, which also leads the period to be divided in two halves: The first half of the decade was dominated by World War II, the widest and most...

Basic idea and motivation

Chern classes are a characteristic class. As such, they are topological invariants associated to vector bundles on a smooth manifold, which means that if you describe the same vector bundle on a manifold in two different ways, the Chern classes will be unaffected. When are two ostensibly different vector bundles the same? When are they different? These questions can be quite hard to answer. But the Chern classes provide a simple test: if the Chern classes of a pair of vector bundles do not agree, then the vector bundles are different. The converse is not true, however.

In topology, differential geometry, and algebraic geometry, it is often of vital importance to know how many linearly independent sections a vector bundle has. The Chern classes can be used to provide a great deal of information in answering this question. For instance, the Riemann-Roch theorem and the Atiyah-Singer index theorem are two theorems which can assist in providing answers to this question, and both involve the Chern class in some form or another. In mathematics, specifically in complex analysis and algebraic geometry, the Riemann-Roch theorem is an important tool in the computation of the dimension of the space of meromorphic functions with prescribed zeroes and allowed poles. ... In the mathematics of manifolds and differential operators, the Atiyah-Singer index theorem is an important unifying result that connects topology and analysis. ...

So Chern classes are useful. But a remarkable fact about them is that they are feasibly calculated. In differential geometry (and many flavors of algebraic geometry), the Chern classes can be expressed as polynomials in the coefficients of the curvature form. For this reason, Chern classes are often at the front lines in attacking many difficult mathematical problems. In differential geometry, the curvature form describes curvature of principal bundle with connection. ...

Example: Hermitian vector bundles on a smooth manifold

Given a complex hermitian vector bundle V of complex rank n over a smooth manifold M, a representative of each Chern class (also called a Chern form) c_k(V) of V are given as the coefficients of the characteristic polynomial of the curvature form Ω of V. A hermitian metric on a complex vector bundle E over a smooth manifold M, is a positive-definite, hermitian inner product on each fiber Ep, that varies smoothly with the point p in M. An important special case is that of a hermitian metric on the complexified tangent bundle of... In mathematics, a vector bundle is a geometrical construct where to every point of a topological space (or manifold, or algebraic variety) we attach a vector space in a compatible way, so that all those vector spaces, glued together, form another topological space (or manifold or variety). ... In mathematics, a vector bundle is a geometrical construct where to every point of a topological space (or manifold, or algebraic variety) we attach a vector space in a compatible way, so that all those vector spaces, glued together, form another topological space (or manifold or variety). ... In mathematics, a manifold M is a type of space, characterized in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. ... In linear algebra, one associates a polynomial to every square matrix, its characteristic polynomial. ... In differential geometry, the curvature form describes curvature of principal bundle with connection. ...

The determinant is over the ring of n×n matrices whose entries are polynomials in t with coefficients in the commutative algebra of even complex differential forms on M. The curvature form Ω of V is defined as In differential geometry, the curvature form describes curvature of principal bundle with connection. ...

with ω the connection and d the exterior derivative, or via the same expression in which ω is a gauge form for the gauge group of V. The scalar t is used here only as an indeterminate to generate the sum from the determinant, and I denotes the n×n identity matrix. In differential geometry, a connection (also connexion) or covariant derivative is a way of specifying a derivative of a vector field along another vector field on a manifold. ... In mathematics, the exterior derivative operator of differential topology, extends the concept of the differential of a function to differential forms of higher degree. ... Gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ... See: indeterminate (variable) statically indeterminate Division by zero This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ... Jump to: navigation, search In mathematics a generating function is a formal power series whose coefficients encode information about a sequence an that is indexed by the natural numbers. ... In linear algebra, the identity matrix of size n is the n-by-n square matrix with ones on the main diagonal and zeros elsewhere. ...

To say that the expression given is a representative of the Chern class indicates that 'class' here means up to addition of an exact differential form. That is, Chern classes are cohomology classes in the sense of de Rham cohomology. Look up Up to on Wiktionary, the free dictionary In mathematics, the phrase up to xxxx indicates that members of an equivalence class are to be regarded as a single entity for some purpose. ... In mathematics, both in vector calculus and in differential topology, the concepts of closed form and exact form are defined for differential forms, by the equations dα = 0 for a given form α to be a closed form, and α = dβ for an exact form, with α given and β... In mathematics, de Rham cohomology is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapted to computation and the concrete representation of cohomology classes. ...

Properties of Chern classes

Given a complex vector bundle V over a topological space X, the Chern classes of V are a sequence of elements of the cohomology of X. The kth Chern class of V, which is usually denoted c_k(V), is an element of In mathematics, a vector bundle is a geometrical construct where to every point of a topological space (or manifold, or algebraic variety) we attach a vector space in a compatible way, so that all those vector spaces, glued together, form another topological space (or manifold or variety). ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a cochain complex. ...

H^2k(X;Z),

the cohomology of X with integer coefficients. One can also define the total Chern class Jump to: navigation, search The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ...

The Chern classes satisfy the following four properties.

• c₀(V) = 1 for all V.

• Functoriality: If is continuous and f ^* V is the pullback of V, then c_k(f ^* V) = f ^* c_k(V).

• Whitney sum formula: If is another complex vector bundle, then the Chern classes of the direct sum are given by

that is, In topology and related areas of mathematics a continuous function is a morphism between topological spaces; that is, a mapping which preserves the topological structure. ... This article discusses the pullback in differential geometry. ... Hassler Whitney (23 March 1907 – 10 May 1989) was an American mathematician who was one of the founders of singularity theory, PhB, Yale University, 1928; MusB, 1929; ScD (Honorary), 1947; PhD, Harvard University, under G.D. Birkhoff, 1932. ... In abstract algebra, the direct sum is a construction which combines several vector spaces (or groups, or abelian groups, or modules) into a new, bigger one. ...

• The total Chern class of the tautological line bundle over is 1 − H, where H is Poincaré-dual to the hyperplane .

In fact, these properties uniquely characterize the Chern classes. They imply, among other things: The canonical or tautological line bundle on a projective space appears frequently in mathematics, often in the study of characteristic classes. ... In mathematics, the Poincaré duality theorem is a basic result on the structure of the homology and cohomology groups of manifolds. ... In geometry, a hyperplane is a linear, affine, or projective subspace of codimension 1. ...

• If n is the complex rank of V, then c_k(V) = 0 for all k > n. Thus the total Chern class terminates.

• The top Chern class of V (meaning c_n(V), where n is the rank of V) is always equal to the Euler class of the underlying real vector bundle.

Since the values are in integral cohomology groups, rather than cohomology with real coefficients, these Chern classes are slightly more refined than those in the Riemannian example. Jump to: navigation, search In mathematics, specifically in algebraic topology, the Euler class is a characteristic class of oriented, real vector bundles. ...

Construction of Chern classes

There are various ways of approaching the subject. Originally Chern used differential geometry. In algebraic topology the Chern classes arise via homotopy theory which provides a mapping associated to V to a classifying space (an infinitary Grassmannian in this case). There is an approach of Alexander Grothendieck showing that axiomatically one need only define the line bundle case. Chern classes also arise naturally in algebraic geometry. An illustration of a homotopy between the two bold paths In topology, two continuous functions from one topological space to another are called homotopic (Greek homeos = identical and topos = place) if one can be continuously deformed into the other, such a deformation being called a homotopy between the two functions. ... In mathematics, a classifying space in homotopy theory of a discrete group G is, roughly speaking, a path connected topological space X such that the fundamental group of X is isomorphic to G and the higher homotopy groups of X are trivial. ... In mathematics, a Grassmannian is the space of all k-dimensional subspaces of an n-dimensional vector space V, often denoted Gk(V) or simply Gk,n. ... Alexander Grothendieck (born March 28, 1928, Berlin) was one of the most important mathematicians active in the 20th century. ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ...

The intuitive meaning of the Chern class concerns 'required zeroes' of a section of a vector bundle: for example the theorem saying one can't comb a hairy ball flat (hairy ball theorem), though that is strictly speaking a question about a real vector bundle. Section can be: A cross section (in the common sense or the physics sense) In mathematics: A conic section A section of a fiber bundle or sheaf A Caesarean section In UK law, Section 28 In the fictional Star Trek universe, Section 31 A military unit A section (land) is... The hairy ball theorem of algebraic topology states that, in laymans terms, one cannot comb the hair on a ball in a smooth manner. This fact is immediately convincing to most people, even though they might not recognize the more formal statement of the theorem: there is no nonvanishing...

See Chern-Simons for more discussion. In mathematics, the Chern-Simons forms are certain secondary characteristic classes. ...

Chern classes of line bundles

If V is a line bundle there is just a single (first) Chern class in the second cohomology group of X. The first Chern class turns out to be a complete invariant with which to classify complex line bundles, topologically speaking. That is, there is a bijection between the isomorphism classes of line bundles over X and the elements of H²(X;Z), which associates to a line bundle its first Chern class. Addition in the second dimensional cohomology group coincides with tensor product of complex line bundles. In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. ... In mathematics, a bijection, bijective function, or one-to-one correspondence is a function that is both injective (one-to-one) and surjective (onto), and therefore bijections are also called one-to-one and onto. ... In mathematics, the tensor product, denoted by , may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras and modules. ...

In algebraic geometry, this classification of (isomorphism classes of) complex line bundles by the first Chern class is a crude approximation to the classification of (isomorphism classes of) holomorphic line bundles by linear equivalence classes of divisors. In mathematics, the concept of a linear system of divisors arose first in the form of a linear system of algebraic curves in the projective plane. ... In mathematics, a divisor of an integer n, also called a factor of n, is an integer which evenly divides n without leaving a remainder. ...

For complex vector bundles of dimension greater than one, the Chern classes are not a complete invariant.

Chern classes of almost complex manifolds and cobordism

The theory of Chern classes gives rise to cobordism invariants for almost complex manifolds. In mathematics, cobordism is a relation between manifolds, based on the idea of boundary. ... In mathematics, an almost complex manifold is a smooth manifold equipped with smooth linear complex structure on each tangent space. ...

If M is an almost complex manifold, then its tangent bundle is a complex vector bundle. The Chern classes of M are thus defined to be the Chern classes of its tangent bundle. If M is also compact and of dimension 2d, then each monomial of total degree 2d in the Chern classes can be paired with the fundamental class of M, giving an integer, a Chern number of M. If M′ is another almost complex manifold of the same dimension, then it is cobordant to M if and only if the Chern numbers of M′ coincide with those of M. In mathematics, the tangent bundle of a differentiable manifold is a vector bundle which as a set is the disjoint union of all the tangent spaces at every point in the manifold with natural topology and smooth structure. ... Jump to: navigation, search Compact as a general noun can refer to: Look up Compact on Wiktionary, the free dictionary a diplomatic contract or covenant among parties, sometimes known as a pact, treaty, or an interstate compact; a British term for a newspaper format; In mathematics, it can refer to... In mathematics, a monomial is a particular kind of polynomial, having just one term. ... In mathematics, the fundamental class is a homology class [M] associated to a manifold M. It is defined (firstly) in cases when M is a closed manifold of dimension n, and oriented. ...

Generalizations

There is a generalization of the theory of Chern classes, where ordinary cohomology is replaced with a generalized cohomology theory. The theories for which such generalization is possible are called complex orientable. The formal properties of the Chern classes remain the same, with one crucial difference: the rule which computes the first Chern class of a tensor product of line bundles in terms of first Chern classes of the factors is not (ordinary) addition, but rather a formal group law. In mathematics, homology theory is the axiomatic study of the intuitive geometric idea of homology of cycles on topological spaces. ... In mathematics, a formal group law is (roughly speaking) the formal power series analogue of a Lie group. ...

See also

• Stiefel-Whitney class

Stiefel-Whitney classes arise in mathematics as a type of characteristic class associated to real vector bundles . ...

References

• Jurgen Jost, Riemannian Geometry and Geometric Analysis, (2002) Springer-Verlag, Berlin ISBN 3-540-4267-2 (Provides a very short, introductory review of Chern classes for Riemannian manifolds).