On the concept of orientability for Fredholm maps between real Banach manifolds

Abstract

In [1] we introduced a concept of orientation and topological degree for nonlinear Fredholm maps between real Banach manifolds. In this paper we study properties of this notion of orientation and we compare it with related results due to Elworthy-Tromba and Fitzpatrick-Pejsachowicz-Rabier.

Orientation and Degree for Fredholm Maps of Index Zero Between Banach Spaces

“…The concept of orientation introduced by Fitzpatrick, Pejsachowicz and Rabier has interesting similarities and differences with our one and the reader can find a comparison (also with the work of Elworthy and Tromba) in [2] and [3].…”

“…We say that A is equivalent to B or, more precisely, A is L-equivalent to B, if det (L + B) −1 (L + A) > 0. This is actually an equivalence relations on C(L), with just two equivalence classes (see [2]). We can therefore introduce the following definition.…”

“…One could actually show that, given a continuous family of Fredholm maps of index zero H : Ω × [0, 1] → F , if H 0 is orientable, then all the partial maps H t are orientable as well, and an orientation of H 0 induces a unique orientation on any H t which makes H oriented (see [3]). …”

“…The properties of this concept of orientation are treated in depth in [2] and [3]. Here we limit ourselves to some remarks.…”

“…C 1 maps such that at any point of the domain the derivative is Fredholm of index zero). This is a revised and simplified version of a recent study in collaboration with Prof. M. Furi (see [2] and [3]). …”