Whitney sum formula - Imported Knowlpedia Content

Statement

Let $E$ and $F$ be vector bundles over the same paracompact base $X$ . The Whitney sum formula states that total characteristic classes multiply under the direct sum:

c(E\oplus F)=c(E)\smile c(F)

for complex bundles and total Chern classes, and

w(E\oplus F)=w(E)\smile w(F)

for real bundles and total Stiefel–Whitney classes. The first identity uses integral cohomology and the second uses coefficients in $\mathbb Z/2$ . In both cases the multiplication is the cup product.

Component formulas

Comparing homogeneous terms gives

c_k(E\oplus F)=\sum_{i+j=k}c_i(E)\smile c_j(F), \qquad w_k(E\oplus F)=\sum_{i+j=k}w_i(E)\smile w_j(F).

Thus $c_1(E\oplus F)=c_1(E)+c_1(F)$ , while higher components include mixed products. The formulas are the product axioms in the standard characterization of these classes; see Milnor–Stasheff, chapter 4 and §14.4.

Examples and consequences

Since $E\oplus\underline{\mathbb F}^{\,m}$ has the same total characteristic class as $E$ , stable isomorphism preserves Chern, Stiefel–Whitney, and Pontryagin classes. The stable triviality

TS^n\oplus\underline{\mathbb R}\cong\underline{\mathbb R}^{\,n+1}

therefore forces all positive-degree Stiefel–Whitney classes of $TS^n$ to vanish.

References

John W. Milnor and James D. Stasheff, Characteristic Classes, Princeton University Press, 1974. DOI record. Relevant: chapter 4, the Stiefel–Whitney product axiom; §14.4, the product theorem for Chern classes.
Dale Husemoller, Fibre Bundles, 3rd ed., Springer, 1994. DOI record. Relevant: the part on characteristic classes and Whitney sums.