Topological K-theory and symbol classes

For a compact space X, complex K⁰(X) is the Grothendieck group of finite-rank complex vector bundles under direct sum. A virtual class [E]−[F] retains more than rank: two equally ranked bundles can have different topology. For a locally compact space Y, compactly supported K-theory is the reduced K-group of its one-point compactification. It can be represented by triples of bundles with an isomorphism outside a compact set, modulo stabilization and homotopy.

For a closed manifold M, pull the source and target bundles of an elliptic operator back to T*M. Its principal symbol identifies them away from the zero section. Restricting to a cotangent disk bundle gives a relative class in K⁰(D*M,S*M), naturally K⁰_c(T*M). The compact zero section contains the noninvertibility locus. The symbol usually grows or stays nonzero at infinity; setting it to zero there would destroy the isomorphism required by the triple.

Complex Bott periodicity identifies K⁰_c(R²) with Z and, more generally, K⁰_c(R^(2N)) with Z. Fix the positive generator β on the oriented complex plane by its compactly supported Chern character integral equal to one. Equivalently, its clutching description is a line bundle of first Chern number one minus a trivial line on the compactification sphere. Declaring this normalization avoids confusing the Hopf line, its dual, or opposite clutching orientations.

A scalar map z raised to a positive integer is invertible off zero and winds that many times on the positively oriented unit circle. For our source-to-target triple, glue the source on the inside disk to the target outside by v_out=σv_in. This gives the negative-degree line for positive winding, so its class is the negative of that multiple of β. Conjugation reverses the sign, and direct sum adds classes. These relative models do not specify a global differential operator.

Embed M in a sufficiently large Euclidean space and use a metric to identify tangent and cotangent descriptions. The induced embedding of total tangent bundles has normal bundle formed from two copies of the original normal bundle. Pairing those copies gives it a complex structure, so a complex K-theory Thom class defines a pushforward. Extension through a tubular neighborhood lands in compactly supported K-theory of an even-dimensional Euclidean space. Inverting Bott periodicity then produces an integer, the topological index.

This construction works for the symbol of a general elliptic operator without assuming that M itself is spin. A direct pushforward from K⁰(M) to a point would require appropriate K-orientation data, often spin-c; the cotangent-symbol construction supplies its own relevant normal orientation. The Chern character translates K-theory into even rational cohomology and becomes an isomorphism after tensoring with Q for finite CW spaces. It can lose torsion information before that rationalization, so a characteristic-form calculation does not describe every integral family invariant.