topological K-theory
/ KAY-theory /
Bundles over a space X can be added (direct sum) but they cannot be subtracted — you cannot literally take E minus E'. K-theory is the algebraic move that forces subtraction to exist, the same trick that builds the integers out of the natural numbers by allowing formal differences. The result, K(X), is a ring built so cleanly from the vector bundles over X that bundle questions become questions of ordinary algebra and even of generalized cohomology.
Start with the set of isomorphism classes of complex vector bundles over X. Direct sum makes this a commutative monoid (you can add, with the zero bundle as identity, but not subtract). The Grothendieck group construction formally adjoins inverses: K(X) consists of formal differences [E] - [F] of bundle classes, where [E] - [F] = [E'] - [F'] precisely when E + F' + G is isomorphic to E' + F + G for some bundle G (the extra G absorbs stabilization). Tensor product of bundles makes K(X) a commutative ring, with the trivial line bundle as multiplicative identity. There is a real version KO(X) built from real bundles, and a reduced version that strips off the trivial part.
K-theory is a genuine generalized cohomology theory: it is homotopy invariant, has the right long exact sequences, and satisfies all the Eilenberg-Steenrod axioms except the dimension axiom (K of a point is Z, not concentrated in one degree). Its signature feature is Bott periodicity — complex K-theory repeats every 2 degrees, real every 8 — which makes it spectacularly computable. It is the natural home of the Atiyah-Singer index theorem, and via the Chern character it is rationally just even-degree ordinary cohomology, but integrally it sees more, including torsion phenomena ordinary cohomology organizes differently.
For a point, K(point) = Z: a bundle over a point is just a vector space, and its class is its dimension, with formal differences giving all of Z. For the 2-sphere, K(S^2) = Z + Z (rank 2 over Z): one Z is the rank, the other is generated by the class [H] - 1 where H is the tautological line bundle of CP^1 = S^2. This second generator detects the nontrivial twisting that ordinary dimension cannot see.
K(point) = Z counts dimension; K(S^2) = Z + Z, the extra factor coming from the twisted tautological line bundle.
K-theory fails the dimension axiom on purpose: K of a point is not concentrated in degree 0, which is exactly what makes it a generalized (extraordinary) cohomology theory rather than ordinary cohomology. Do not expect K^n(point) to vanish for n nonzero — Bott periodicity fills the even degrees.