Lefschetz-Riemann-Roch theorem and coherent trace formula.
A homology theory of Banach manifolds of a special form, called FSQL-manifolds, is developed, and also a homological degree of FSQL-mappings between FSQL-manifolds is introduced.
We prove that on a -complex the obstruction for a line bundle to be the fractional power of a suitable pullback of the Hopf bundle on a 2-dimensional sphere is the vanishing of the square of the first Chern class of . On the other hand we show that if one looks at integral powers then further secondary obstructions exist.
We exhibit a six dimensional manifold with a line bundle on it which is not the pullback of a bundle on .
We study secondary obstructions to representing a line bundle as the pull-back of a line bundle on and we interpret them geometrically.