the Chern character
/ CHURN / CHEN /
K-theory and ordinary cohomology are two different bookkeeping systems for bundles. They look very different integrally, but rationally they ought to carry the same information, and one wants a precise dictionary translating one into the other. The Chern character ch is that dictionary: it is a ring homomorphism from K-theory into rational even-degree cohomology that becomes an isomorphism after tensoring with the rationals.
It is defined cleanly by the splitting principle. For a line bundle L with first Chern class x = c_1(L), set ch(L) = exp(x) = 1 + x + x^2/2! + x^3/3! + .... For a general bundle E, pretend by splitting that E = L_1 + ... + L_n with Chern roots x_1, ..., x_n, and define ch(E) = sum of exp(x_i) = n + (sum x_i) + (sum x_i^2)/2 + ... = rank + c_1 + (c_1^2 - 2c_2)/2 + .... Because exp turns sums into products correctly, ch is additive AND multiplicative: ch(E + E') = ch(E) + ch(E') and ch(E tensor E') = ch(E) ch(E'). That is exactly the statement that ch is a ring homomorphism ch: K(X) -> H^even(X; Q), and the theorem is that ch tensored with Q is a ring isomorphism K(X) tensor Q -> H^even(X; Q).
The Chern character is the precise sense in which 'K-theory is rationally just even cohomology'. It is indispensable in the index theorem, where the topological index is assembled from ch of the symbol class together with the Todd class. The caveat to internalize is the word rationally: ch is an isomorphism only after tensoring with Q. Integrally, K-theory carries strictly more information — torsion classes that ch annihilates (it kills all torsion, since H^even(X; Q) has none). So two bundles can have equal Chern character yet be distinct in K-theory.
For a complex line bundle L with c_1(L) = x, the Chern character is ch(L) = 1 + x + x^2/2 + .... If you tensor two line bundles, c_1(L tensor L') = x + x', and indeed ch(L tensor L') = exp(x + x') = exp(x) exp(x') = ch(L) ch(L'). The exponential law for c_1 of a tensor product is exactly the multiplicativity of ch made visible.
Because c_1 of a tensor product adds, and exp turns addition into multiplication, ch is a ring map.
The Chern character is an isomorphism only rationally (after tensoring with Q). It kills all torsion, so integrally K-theory contains information ch cannot see; equal Chern characters do not imply equal K-theory classes.