A vectorial bundle is a -graded vector bundle of finite rank, equipped with an odd endomorphism . Homomorphisms of vectorial bundles are such that the endomorphism acts like canceling parts of the even and odd degree of against each other.
This way vectorial bundles lend themselves to the description of K-theory. In particular, they allow a geometric model for twisted K-theory.
as morphisms equivalence classes of morphisms of vector bundles such that
where two such maps are regarded as equivalent, , already if they coincide on the kernel of for each point .
In particular, we have the following two important special cases:
the case that – in this case all eigenvalues of all are zero. and hence maps represent the same morphism precisely if they are actually equal as morphisms of vector bundles.
(Notice that there is only the 0-morphism for .)
This yields a canonical inclusion
by sending .
the case that and
Here and hence two morphisms are identified already if they agree on the 0-vector. In other words, all morphisms out of such are identified. In particular they are all equal to the 0-morphism to . Therefore the bundles of this form represent the 0-element.
Say two vectorial bundles , on are concordant if there is a vectorial bundle on which restricts to them at either end, respectively.
Let be the degree-reversed bundle to .
There is a concordance
The definition of vectorial bundles is due to Furuta. It is recalled and applied to the study of K-theory and twisted K-theory in
K. Gomi, Twisted K-theory and finite-dimensional approximation (arXiv)
Revised on July 21, 2009 10:30:55
by Urs Schreiber