A multiplicative characteristic-class homomorphism from K-theory to cohomology that assigns to a vector bundle its total Chern character, a rational cohomology class which measures topological charge and linearizes the tensor product in K-theory.

Principle

Translate K-theory classes into cohomological invariants via the exponential of Chern classes so that additive structure in K-theory corresponds to sum in even-degree cohomology and tensor product corresponds (multiplicatively) to cup product after passing to rational coefficients.

Demonstration

For a complex line bundle L over a smooth manifold, ch(L)=exp(c1(L)) in H^{even}(X;Q). For a direct sum E⊕F, ch(E⊕F)=ch(E)+ch(F), and for a tensor product ch(E⊗F)=ch(E)∪ch(F) after rationalization.

Misapplication

Treating the Chern character as an integral isomorphism and ignoring torsion: it generally loses torsion information and is only an isomorphism after tensoring K-theory with Q; treating it as equal to the total Chern class is a category error.

Consequence

Enables index formulas and Riemann–Roch type theorems by transferring K-theory problems to cohomology, where characteristic numbers and integrals compute indices and topological charges.

Reversal

Reconstructing a unique K-theory class from a cohomology class is not canonical: an inverse exists only rationally and is non-unique because integral and torsion data in K-theory are invisible to the Chern character.

Boundary

Applies in topological, differentiable, and algebraic contexts with appropriate cohomology theories but requires rational coefficients to be fully faithful; it does not recover torsion K-theory and must be adjusted on singular spaces or for equivariant or p-adic theories.

Semantic Tension

Close to the total Chern class and to characteristic classes generally, the tension is between multiplicative, additive, and integral vs rational viewpoints: Chern character linearizes multiplicative K-theory structure at the cost of torsion information.

Synthesis

The Chern character is the rational bridge from K-theory to cohomology: it converts vector-bundle data into cohomological invariants via exponentiated Chern classes, enabling computations and index results while deliberately discarding torsion subtleties.