An elliptic complex on a smooth manifold is a finite cochain complex of differential operators between sections of vector bundles,with , whose principal symbol sequence is exact at every nonzero real cotangent vector. Thus for , the kernel of each symbol map equals the image of the preceding one, including injectivity and surjectivity at the ends.[ref]
A Lefschetz Fixed Point Formula for Elliptic Complexes: II. Applications by M. F. Atiyah and R. Bott. section 4, definition of elliptic complexes and leading symbols, printed page 452
The de Rham complex is the basic example. Up to an inessential nonzero scalar convention, the symbol of is . Choose a tangent vector with . The interior product identitygives a contracting homotopy for that symbol sequence and proves exactness.
The condition belongs to the whole sequence: the individual maps need not be elliptic operators between equal-rank bundles. On a compact manifold without boundary, ellipticity gives finite-dimensional cohomology. Boundary problems require additional conditions. This finiteness is what makes an alternating cohomological trace meaningful in the Atiyah-Bott fixed-point theorem.
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An elliptic complex is a concept in the field of mathematics, specifically within the areas of partial differential equations and the theory of elliptic operators. It relates to elliptic differential operators and the mathematical structures associated with them. ### Key Concepts: 1. **Elliptic Operators**: These are a class of differential operators that satisfy a certain condition (the ellipticity condition), which ensures the well-posedness of boundary value problems. An operator is elliptic if its principal symbol is invertible.