Abstract: 

When dealing with the existence of solutions of elliptic partial differential equations, it is useful to have a topological degree for nonlinear Fredholm operators of index zero. In defining this degree, the notion of orientability of mappings h : X ! Φ0(U, V ), where Φ0(U, V ) denotes the space of Fredholm operators of index zero between two real Banach spaces, is imperative. This motivates the study of the set of homotopy classes [X, Φ0(U, V )]. In view of this, in this talk we will study the structure of [X, Φ0(U, V )] by means of the nonlinear spectral theory and more concretely via the generalised algebraic multiplicity. Moreover, it will be seen that this analysis can be rephrased to the language of topological K-theory of vector bundles, allowing to study the orientability of vector bundles by means of infinite dimensional data.