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For two vector bundles, their direct sum over , also called their Whitney sum, is the vector bundle whose fiber over any is the direct sum of vector spaces of the fibers of and (the fiber-wise direct sum).
(direct sum of topological vector bundles via total spaces)
Let
be a topological space,
and two topological vector bundles over .
Then the direct sum of vector bundles is the topological vector bundle whose total space is the topological subspace
of the product topological space of the two total spaces, and whose projection map is
For the vector space structure on the fibers
is the one on the direct sum of vector spaces.
(direct sum of topological vector bundles via transition functions)
Let be a topological space, and let and be two topological vector bundles over .
Let be an open cover with respect to which both vector bundles locally trivialize (this always exists: pick a local trivialization of either bundle and form the joint refinement of the respective open covers by intersection of their patches). Let
be the transition functions of these two bundles with respect to this cover.
For write
be the pointwise direct sum of these transition functions
Then the direct sum bundle is the one glued from this direct sum of the transition functions (by this construction):
Let and be topological spaces, and write for their disjoint union space.
Then every topological vector bundle on is the direct sum of a vector bundle that has rank zero on and one that has rank zero on .
More explicitiy: let
and
be the operations of extending a vector bundle on the other connected component by a rank-0 vector bundle, then
is an isomorphism of isomorphism classes of vector bundles (and an equivalence of categories of categories of vector bundles before passing to isomorphism classes).
(sub-bundles over paracompact spaces are direct summands)
Let
Then every vector subbundle is a direct vector bundle summand, in that there exists another vector subbundle such that their direct sum of vector bundles (def. ) is
Since is assumed to be paracompact Hausdorff, there exists a inner product on vector bundles
(by this prop.). This defines at each the orthogonal complement of . The subspace of these orthogonal complements is readily checked to be a topological vector bundle . Hence by construction we have
(over compact Hausdorff spaces every vector bundle is direct summand of a trivial bundle)
Let
Then there exists another topological vector bundle such that the direct sum of vector bundles (def. ) of the two is a trivial vector :
(e.g. Hatcher, prop. 1.4, Friedlander, ptop. 3.1)
Let be an open cover of over which has a local trivialization
By compactness of , there is a finite sub-cover, hence a finite set such tat
is still an open cover over which trivializes.
Since paracompact Hausdorff spaces equivalently admit subordinate partitions of unity there exists a partition of unity
with support . Hence the functions
extend by 0 to vector bundle homomorphism of the form
The finite pointwise direct sum of these yields a vector bundle homomorphism of the form
Observe that, as opposed to the single , this is a fiber-wise injective, because at each point at least one of the is non-vanishing. Hence this is an injection of into a trivial vector bundle.
Prop. is key for the construction of topological K-theory groups on compact Hausdorff spaces.
Remark : Let and be vector bundles over . This gives product map which is still a vector bundle. Consider diagonal map given by . The Whitney sum of and is the pull back of along the diagonal map which is denoted by .
(Euler class takes Whitney sum to cup product)
The Euler class of the Whitney sum of two oriented real vector bundles to the cup product of the separate Euler classes:
For details see at Euler class, this Prop..
Discussion in a context of topological K-theory:
Max Karoubi, §4.8 in: K-Theory – An introduction, Grundlehren der mathematischen Wissenschaften 226 Springer (1978) [pdf, doi:10.1007%2F978-3-540-79890-3]
Allen Hatcher, section 1.1 of Vector bundles and K-Theory, (partly finished book) web
and with an eye towards algebraic K-theory;
Last revised on May 17, 2023 at 10:06:52. See the history of this page for a list of all contributions to it.