[2][3] The index problem for elliptic differential operators was posed by Israel Gel'fand.
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).
[5] It appears also in the "Séminaire Cartan-Schwartz 1963/64"[6] that was held in Paris simultaneously with the seminar led by Richard Palais at Princeton University.
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.
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).
is given by in other words the value of the top dimensional component of the mixed cohomology class
, then applying the Thom isomorphism and dividing by the Euler class,[26][27] the topological index may be expressed as where division makes sense by pulling
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).
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.
Although the analytical index is usually hard to evaluate directly, it is at least obviously an integer.
Using topological cobordism and K-homology one may provide a full statement of an index theorem on quasiconformal manifolds (see page 678 of (Connes, Sullivan & Teleman 1994)).
At the same time, they provide, also, an effective construction of the rational Pontrjagin classes on topological manifolds.
The obstruction theories due to Milnor, Kervaire, Kirby, Siebenmann, Sullivan, Donaldson show that only a minority of topological manifolds possess differentiable structures and these are not necessarily unique.
, in order to prove assertions involving characteristic classes, we may suppose that there are complex line bundles
Take X to be a complex manifold of (complex) dimension n with a holomorphic vector bundle V. We let the vector bundles E and F be the sums of the bundles of differential forms with coefficients in V of type (0, i) with i even or odd, and we let the differential operator D be the sum restricted to E. This derivation of the Hirzebruch–Riemann–Roch theorem is more natural if we use the index theorem for elliptic complexes rather than elliptic operators.
Then the i'th cohomology group is just the coherent cohomology group Hi(X, V), so the analytical index of this complex is the holomorphic Euler characteristic of V: Since we are dealing with complex bundles, the computation of the topological index is simpler.
Using Chern roots and doing similar computations as in the previous example, the Euler class is given by
The operator D is the Hodge Laplacian restricted to E, where d is the Cartan exterior derivative and d* is its adjoint.
The  genus is a rational number defined for any manifold, but is in general not an integer.
Borel and Hirzebruch showed that it is integral for spin manifolds, and an even integer if in addition the dimension is 4 mod 8.
In this case, constant coefficient differential operators are just the Fourier transforms of multiplication by polynomials, and constant coefficient pseudodifferential operators are just the Fourier transforms of multiplication by more general functions.
Also, there is a direct correspondence between data representing elements of K(B(X), S(X)) (clutching functions) and symbols of elliptic pseudodifferential operators.
Pseudodifferential operators have an order, which can be any real number or even −∞, and have symbols (which are no longer polynomials on the cotangent space), and elliptic differential operators are those whose symbols are invertible for sufficiently large cotangent vectors.
The initial proof was based on that of the Hirzebruch–Riemann–Roch theorem (1954), and involved cobordism theory and pseudodifferential operators.
Atiyah, Bott, and Patodi (1973) gave a new proof of the index theorem using the heat equation, see e.g. Berline, Getzler & Vergne (1992).
Therefore, the index of D is given by for any positive t. The right hand side is given by the trace of the difference of the kernels of two heat operators.
These have an asymptotic expansion for small positive t, which can be used to evaluate the limit as t tends to 0, giving a proof of the Atiyah–Singer index theorem.