In differential geometry, the Atiyah–Singer index theorem, proved by Michael Atiyah and Isadore Singer (1963), states that for an elliptic differential operator on a compact manifold, the analytical index (related to the dimension of the space of solutions) is equal to the topological index (defined in terms of some topological data). It includes many other theorems, such as the Chern–Gauss–Bonnet theorem and Riemann–Roch theorem, as special cases, and has applications to theoretical physics.
History The index problem for elliptic differential operators was posed by Israel Gel'fand. He noticed the homotopy invariance of the index, and asked for a formula for it by means of topological invariants. Some of the motivating examples included the Riemann–Roch theorem and its generalization the Hirzebruch–Riemann–Roch theorem, and the Hirzebruch signature theorem. Friedrich Hirzebruch and Armand Borel had proved the integrality of the  genus of a spin manifold, and Atiyah suggested that this integrality could be explained if it were the index of the Dirac operator (which was rediscovered by Atiyah and Singer in 1961).
The Atiyah–Singer theorem was announced in 1963. The proof sketched in this announcement was never published by them, though it appears in Palais's book. It appears also in the "Séminaire Cartan-Schwartz 1963/64" that was held in Paris simultaneously with the seminar led by Richard Palais at Princeton University. The last talk in Paris was by Atiyah on manifolds with boundary. Their first published proof replaced the cobordism theory of the first proof with K-theory, and they used this to give proofs of various generalizations in another sequence of papers.
1965: Sergey P. Novikov published his results on the topological invariance of the rational Pontryagin classes on smooth manifolds. Robion Kirby and Laurent C. Siebenmann's results, combined with René Thom's paper proved the existence of rational Pontryagin classes on topological manifolds. The rational Pontryagin classes are essential ingredients of the index theorem on smooth and topological manifolds. 1969: Michael Atiyah defines abstract elliptic operators on arbitrary metric spaces. Abstract elliptic operators became protagonists in Kasparov's theory and Connes's noncommutative differential geometry. 1971: Isadore Singer proposes a comprehensive program for future extensions of index theory. 1972: Gennadi G. Kasparov publishes his work on the realization of K-homology by abstract elliptic operators. 1973: Atiyah, Raoul Bott, and Vijay Patodi gave a new proof of the index theorem using the heat equation, described in a paper by Melrose. 1977: Dennis Sullivan establishes his theorem on the existence and uniqueness of Lipschitz and quasiconformal structures on topological manifolds of dimension different from 4. 1983: Ezra Getzler motivated by ideas of Edward Witten and Luis Alvarez-Gaume, gave a short proof of the local index theorem for operators that are locally Dirac operators; this covers many of the useful cases. 1983: Nicolae Teleman proves that the analytical indices of signature operators with values in vector bundles are topological invariants. 1984: Teleman establishes the index theorem on topological manifolds. 1986: Alain Connes publishes his fundamental paper on noncommutative geometry. 1989: Simon K. Donaldson and Sullivan study Yang–Mills theory on quasiconformal manifolds of dimension 4. They introduce the signature operator S defined on differential forms of degree two. 1990: Connes and Henri Moscovici prove the local index formula in the context of non-commutative geometry. 1994: Connes, Sullivan, and Teleman prove the index theorem for signature operators on quasiconformal manifolds.
Notation X is a compact smooth manifold (without boundary). E and F are smooth vector bundles over X. D is an elliptic differential operator from E to F. So in local coordinates it acts as a differential operator, taking smooth sections of E to smooth sections of F.
Symbol of a differential operator If D is a differential operator on a Euclidean space of order n in k variables , then its symbol is the function of 2k variables , given by dropping all terms of order less than n and replacing by . So the symbol is homogeneous in the variables y, of degree n. The symbol is well defined even though does not commute with because we keep only the highest order terms and differential operators commute "up to lower-order terms". The operator is called elliptic if the symbol is nonzero whenever at least one y is nonzero.
Example: The Laplace operator in k variables has symbol , and so is elliptic as this is nonzero whenever any of the 's are nonzero. The wave operator has symbol , which is not elliptic if , as the symbol vanishes for some non-zero values of the ys.
The symbol of a differential operator of order n on a smooth manifold X is defined in much the same way using local coordinate charts, and is a function on the cotangent bundle of X, homogeneous of degree n on each cotangent space. (In general, differential operators transform in a rather complicated way under coordinate transforms (see jet bundle); however, the highest order terms transform like tensors so we get well defined homogeneous functions on the cotangent spaces that are independent of the choice of local charts.) More generally, the symbol of a differential operator between two vector bundles E and F is a section of the pullback of the bundle Hom(E, F) to the cotangent space of X. The differential operator is called elliptic if the element of Hom(Ex, Fx) is invertible for all non-zero cotangent vectors at any point x of X.
A key property of elliptic operators is that they are almost invertible; this is closely related to the fact that their symbols are almost invertible. More precisely, an elliptic operator D on a compact manifold has a (non-unique) parametrix (or pseudoinverse) D′ such that DD′ -1 and D′D -1 are both compact operators. An important consequence is that the kernel of D is finite-dimensional, because all eigenspaces of compact operators, other than the kernel, are finite-dimensional. (The pseudoinverse of an elliptic differential operator is almost never a differential operator. However, it is an elliptic pseudodifferential operator.)
Analytical index As the elliptic differential operator D has a pseudoinverse, it is a Fredholm operator. Any Fredholm operator has an index, defined as the difference between the (finite) dimension of the kernel of D (solutions of Df = 0), and the (finite) dimension of the cokernel of D (the constraints on the right-hand-side of an inhomogeneous equation like Df = g, or equivalently the kernel of the adjoint operator). In other words, Index(D) = dim Ker(D) − dim Coker(D) = dim Ker(D) − dim Ker(D*). This is sometimes called the analytical index of D.
Example: Suppose that the manifold is the circle (thought of as R/Z), and D is the operator d/dx − λ for some complex constant λ. (This is the simplest example of an elliptic operator.) Then the kernel is the space of multiples of exp(λx) if λ is an integral multiple of 2πi and is 0 otherwise, and the kernel of the adjoint is a similar space with λ replaced by its complex conjugate. So D has index 0. This example shows that the kernel and cokernel of elliptic operators can jump discontinuously as the elliptic operator varies, so there is no nice formula for their dimensions in terms of continuous topological data. However the jumps in the dimensions of the kernel and cokernel are the same, so the index, given by the difference of their dimensions, does indeed vary continuously, and can be given in terms of topological data by the index theorem.
Topological index The topological index of an elliptic differential operator between smooth vector bundles and on an -dimensional compact manifold is given by
in other words the value of the top dimensional component of the mixed cohomology class on the fundamental homology class of the manifold up to a difference of sign. Here,
is the Todd class of the complexified tangent bundle of . is equal to , where is the Thom isomorphism for the sphere bundle is the Chern character is the "difference element" in associated to two vector bundles and on and an isomorphism between them on the subspace . is the symbol of
In some situations, it is possible to simplify the above formula for computational purposes. In particular, if is a -dimensional orientable (compact) manifold with non-zero Euler class , then applying the Thom isomorphism and dividing by the Euler class, the topological index may be expressed as
where division makes sense by pulling back from the cohomology ring of the classifying space .
One can also define the topological index using only K-theory (and this alternative definition is compatible in a certain sense with the Chern-character construction above). If X is a compact submanifold of a manifold Y then there is a pushforward (or "shriek") map from K(TX) to K(TY). The topological index of an element of K(TX) is defined to be the image of this operation with Y some Euclidean space, for which K(TY) can be naturally identified with the integers Z (as a consequence of Bott-periodicity). This map is independent of the embedding of X in Euclidean space. Now a differential operator as above naturally defines an element of K(TX), and the image in Z under this map "is" the topological index.
As usual, D is an elliptic differential operator between vector bundles E and F over a compact manifold X.
The index problem is the following: compute the (analytical) index of D using only the symbol s and topological data derived from the manifold and the vector bundle. The Atiyah–Singer index theorem solves this problem, and states:
The analytical index of D is equal to its topological index.
In spite of its formidable definition, the topological index is usually straightforward to evaluate explicitly. So this makes it possible to evaluate the analytical index. (The cokernel and kernel of an elliptic operator are in general extremely hard to evaluate individually; the index theorem shows that we can usually at least evaluate their difference.) Many important invariants of a manifold (such as the signature) can be given as the index of suitable differential operators, so the index theorem allows us to evaluate these invariants in terms of topological data.
Although the analytical index is usually hard to evaluate directly, it is at least obviously an integer. The topological index is by definition a rational number, but it is usually not at all obvious from the definition that it is also integral. So the Atiyah–Singer index theorem implies some deep integrality properties, as it implies that the topological index is integral.
The index of an elliptic differential operator obviously vanishes if the operator is self adjoint. It also vanishes if the manifold X has odd dimension, though there are pseudodifferential elliptic operators whose index does not vanish in odd dimensions.
Relation to Grothendieck–Riemann–Roch The Grothendieck–Riemann–Roch theorem was one of the main motivations behind the index theorem because the index theorem is the counterpart of this theorem in the setting of real manifolds. Now, if there's a map of compact stably almost complex manifolds, then there is a commutative diagram
if is a point, then we recover the statement above. Here is the Grothendieck group of complex vector bundles. This commutative diagram is formally very similar to the GRR theorem because the cohomology groups on the right are replaced by the Chow ring of a smooth variety, and the Grothendieck group on the left is given by the Grothendieck group of algebraic vector bundles.
Extensions of the Atiyah–Singer index theorem
Teleman index theorem Due to , :
For any abstract elliptic operator on a closed, oriented, topological manifold, the analytical index equals the topological index.