The purpose of this paper is to
show that even though vector bundles cannot in general be cancelled from direct
sums (Whitney sums), in certain low-rank situations vector bundles can be
cancelled at the expense of complexifying or quaternionifying the remaining
terms. To be specific, let λ, ξ1, ξ2 be vector bundles over a paracompact
space X, such that λ ⊕ ξ1≅λ ⊕ ξ2. First assume that these are real vector
bundles. If λ is a line bundle of finite type, then the complexifications of ξ1 and
ξ2 are isomorphic, and hence 2ξ1≅2ξ2 (where 2ξi denotes the direct sum
of two copies of ξi), while if λ is a direct sum of two line bundles of finite
type, then the quaternionifications of ξ1 and ξ2 are isomorphic, and hence
4ξ1≅4ξ2. Now assume that these are complex vector bundles. If λ is the
complexification of a real tine bundle of finite type (in particular, λ could be a trivial
complex vector bundle of rank 1), then the quaternionifications of ξ1 and ξ2
are isomorphic, and hence ξ1⊕ξ1≅ξ2⊕ξ2 (where ξi denotes the conjugate
vector bundle to ξi). These results are independent of the dimension of the
space X, and also independent of the dimensions of the fibres of ξ1 and
ξ2. The same results also hold for smooth vector bundles over a smooth
manifold.
