the Whitney sum formula
/ WHIT-nee /
When you stack two bundles on top of each other — taking their direct sum, the Whitney sum, which glues two vector spaces fiberwise into one larger one — how do their characteristic classes combine? The Whitney sum formula is the clean multiplication rule: the total characteristic class of a sum is the product of the totals. It is the bookkeeping principle that lets you compute the classes of complicated bundles by decomposing them into simpler pieces.
Precisely, for the total Chern class of complex bundles E and F over the same base, c(E (+) F) = c(E) c(F), where the product is the cup product in cohomology and the total class is c = 1 + c_1 + c_2 + .... Reading off each degree: c_1(E (+) F) = c_1(E) + c_1(F), while c_2(E (+) F) = c_2(E) + c_1(E) c_1(F) + c_2(F), and so on — the individual classes are NOT additive, only the totals multiply. The same statement holds for total Stiefel-Whitney classes (w(E (+) F) = w(E) w(F)) and, modulo 2-torsion, for total Pontryagin classes. From the Chern-Weil side it is transparent: the curvature of a direct-sum connection is block-diagonal, so det(I + (i/2pi) Omega) factors as the product of the two blocks' determinants.
Combined with the splitting principle — the trick that lets you pretend any bundle is a direct sum of line bundles for the purpose of computing characteristic classes — the Whitney sum formula is the computational engine of the whole theory. You write the total class of a sum of line bundles as a product of factors (1 + x_i), where the x_i are the first Chern classes of the line summands (the Chern roots), and then every characteristic class becomes a symmetric polynomial in the x_i. One caveat to remember: the formula is multiplicative for the TOTAL class, so it is the individual classes in each degree that carry cross terms; misremembering it as 'classes add' is a common error that gives wrong answers from c_2 onward.
Take two complex line bundles L and M with c_1(L) = x and c_1(M) = y. Their Whitney sum L (+) M is a rank-2 bundle, and c(L (+) M) = (1 + x)(1 + y) = 1 + (x + y) + xy. So c_1(L (+) M) = x + y (the classes add in degree 2) but c_2(L (+) M) = xy — a product, not a sum. This tiny computation, scaled up by the splitting principle, is how one derives every formula relating Chern classes of sums, tensor products, and duals.
Total classes multiply: c(L+M) = (1+x)(1+y); the cross term xy is c_2.
It is the TOTAL class that multiplies, not the individual classes. So c_1 happens to add, but c_2 and higher carry cross terms (c_2(E+F) = c_2(E) + c_1(E)c_1(F) + c_2(F)). The formula needs the sum to be a genuine direct (Whitney) sum; it does not apply to tensor products, which obey a different (more intricate) rule.