The Atiyah–Singer index formula

For an elliptic differential operator between smooth complex bundles on a smooth compact manifold without boundary, the analytic index of its Sobolev realization equals the topological index of its class in K⁰_c(T*M). The topological map is the embedding, Thom and Bott construction of the previous lesson. No spin assumption is needed for this general statement. A spin assumption enters only when the operator is specialized to the spin Dirac construction and its density is simplified accordingly.

The theorem determines a difference of kernel dimensions, not the separate kernels or a complete eigenvalue list. It is unchanged by lower-order modifications that preserve the symbol class, although the actual solutions may change. Its proof requires substantial elliptic analysis and topology; computing a characteristic integral in one example verifies the consequences and normalization, rather than furnishing a general proof. Boundary or noncompact problems require additional hypotheses and correction terms that this closed-manifold version does not contain.

Use a unitary connection with skew-Hermitian curvature F_E and normalize the Chern character as tr exp(iF_E/(2π)). For the real tangent curvature matrix R, take p₁=−tr(R∧R)/(8π²). These choices give the  expansion starting with one minus p₁ divided by twenty-four. Only the top-degree component of the product is integrated. All lower-degree terms matter through multiplication, but terms above the manifold dimension make no contribution.

In real dimension four, the twisted spin integrand is one half of c₁(E)² minus c₂(E), minus rank(E) times p₁(TM)/24. Characteristic numbers may be supplied topologically or computed from normalized curvature forms. Changing the connection changes the representative by an exact form and leaves the integral unchanged on a closed manifold. The rational coefficients do not make the final answer an arbitrary rational number: the theorem imposes integrality for the stated geometric data.

For the Dolbeault operator on a compact complex manifold, rolled from even to odd antiholomorphic degrees and twisted by a holomorphic bundle E, the index is the integral of Td(T^(1,0)M) times ch(E). This is the holomorphic Euler characteristic. A Kähler metric is convenient for many analytic identities but is not required for this Dolbeault index formula. On a complex curve the Todd class truncates to one plus half the first Chern class of the tangent line.

The de Rham parity operator instead pairs the Euler class with the fundamental class. A spin-c Dirac formula includes the exponential of half its determinant-line class along with Â. These variants explain why an oriented manifold with no spin lift can still have an index theorem for other elliptic operators. They also prevent a common mistake: replacing every occurrence of Todd or Euler by  simply because all three appear in the broader subject of index theory.