The de Rham complex and Euler characteristic

Work with complex-valued differential forms on a smooth closed oriented Riemannian manifold. The exterior derivative squares to zero, so forms make a cochain complex. At a nonzero real covector ξ, its principal symbol is i times exterior multiplication by ξ. Contraction by the metric dual vector supplies a homotopy inverse up to the positive scalar |ξ|². This proves that the symbol sequence is exact away from the zero section, which is the definition of ellipticity for this complex.

The operator d in one degree need not itself be an elliptic map between equal-rank bundles; the exact sequence of all symbol maps is what matters. One can instead roll the complex into D=d+d*, which exchanges even and odd forms. Exterior multiplication minus contraction satisfies the negative Clifford relation, and the principal symbol of D squares to |ξ|². Thus the even-to-odd part has an invertible symbol and belongs to the Fredholm framework already developed.

The Hodge Laplacian is nonnegative, and its quadratic form on a smooth form is the sum of the squared norms of d and d*. Therefore Dω=0 is equivalent to harmonicity, not merely to being closed. On a closed manifold Hodge theory identifies harmonic k-forms with de Rham cohomology in degree k. With the even-to-odd convention, the kernel contains even harmonic forms and the cokernel contains odd harmonic forms, because the odd-to-even part is the adjoint.

Their difference is the Euler characteristic. This is a metric-independent integer, although harmonic representatives, eigenvalues and inner products do depend on the metric. Changing the metric produces isomorphic cohomology spaces, not identical differential forms point by point. On a manifold with boundary the statement must instead specify absolute or relative Hodge boundary conditions; arbitrary unrestricted smooth forms no longer supply the closed-manifold Fredholm realization used here.

For compact manifolds the cohomology groups are finite dimensional, and the Künneth formula over the field of complex numbers computes product Betti numbers by convolution. Equivalently, their Poincaré polynomials multiply. Evaluating that polynomial at minus one proves multiplicativity of the Euler characteristic. No torsion correction is present over this field; integral homology would require a more careful statement involving torsion and tensor products.

The total dimension of the harmonic space is the polynomial evaluated at plus one, which is usually different from the index. For a connected oriented closed surface of genus g, the Betti numbers are one, twice g, and one. A negative Euler characteristic therefore means more odd than even harmonic forms, not a negative number of solutions. Products make that interpretation especially useful because a positive index can result from multiplying two negative alternating counts.