Giovanni Landi
Projective Modules of Finite Type and Monopoles over $S^2$
(45K, latex)
ABSTRACT. We give a unifying description of all inequivalent vector bundles over the
$2$- dimensional sphere $S^2$ by constructing suitable global projectors $p$ via
equivariant maps. Each projector determines the projective module of finite type of
sections of the corresponding complex rank $1$ vector bundle over $S^2$.
The canonical connection $\nabla = p \circ d$ is used to compute the topological
charges. Transposed projectors gives opposite values for the charges, thus showing
that transposition of projectors, although an isomorphism in $K$-theory, is not the
identity map. Also, we construct the partial isometry yielding the equivalence between
the tangent projector (which is trivial in $K$-theory) and the real form of the
charge $2$ projector.