Knowlpedia

Theorem

Whitney sum formula

Total characteristic classes multiply when vector bundles are combined by direct sum.

Statement

Let EE and FF be real or complex topological vector bundles over the same paracompact Hausdorff base XX. The Whitney sum formula states that total multiply under the fiberwise direct sum (with its topology defined by product bundle charts):

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

for complex bundles and , and

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

for real bundles and . The first identity uses and the second uses coefficients in Z/2\mathbb Z/2. In both cases the multiplication is the .

Component formulas

Comparing homogeneous terms gives

ck(E⊕F)=∑i+j=kci(E)⌣cj(F),wk(E⊕F)=∑i+j=kwi(E)⌣wj(F).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 c1(E⊕F)=c1(E)+c1(F)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.

Related characteristic classes

Total satisfy the following identity after passing to rational cohomology:

p(E⊕F)=p(E)⌣p(F).p(E\oplus F)=p(E)\smile p(F).

Integrally the difference can be nonzero 22-torsion. The extra terms come from products of odd Chern classes of the complexifications, each annihilated by 22. Thus one must not assert the integral Pontryagin product formula without a condition eliminating this torsion.

For oriented real bundles, compatible product orientations give

e(E⊕F)=e(E)⌣e(F)e(E\oplus F)=e(E)\smile e(F)

for the . These identities use their own coefficient rings and degree conventions.

Examples and consequences

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

TSn⊕R‾≅R‾ n+1TS^n\oplus\underline{\mathbb R}\cong\underline{\mathbb R}^{\,n+1}

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

References
  1. 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.
  2. Dale Husemoller, Fibre Bundles, 3rd ed., Springer, 1994. DOI record. Relevant: the part on characteristic classes and Whitney sums.