---
tag: BE9NJRVDR3
kind: concept
name: Whitney sum
msc: [55R25, 57R22]
sources:
  - url: https://en.wikipedia.org/wiki/Vector_bundle
---

## Summary

The Whitney sum $E \oplus E'$ of two vector bundles over the same base has as fiber over each point the direct sum of the two fibers.
It is the basic operation on vector bundles, and with it the isomorphism classes of vector bundles over a space form a commutative monoid.
[[Characteristic classes|7HACJC0JE2]] behave multiplicatively under Whitney sum, which is the content of the Whitney product formula.
Over a compact Hausdorff space, topological K-theory is the group completion of this monoid.

## Explanations

### L1 human

Over each point of a base space, a vector bundle provides a vector space.
Given two bundles over the same base, place the two vector spaces side by side at each point to form their direct sum.
The result is a new bundle whose rank is the sum of the two ranks.
For the sphere, the tangent planes together with the normal lines give, at each point, the whole of $\R^3$.