On the product formula for the oriented degree for Fredholm maps of index zero between Banach manifolds

Global persistence of the unit eigenvectors of perturbed eigenvalue problems in Hilbert spaces: the odd multiplicity case

“…Instead, we use a notion of topological degree for oriented Fredholm maps of index zero between real differentiable Banach manifolds, developed by two authors of this paper, and whose construction and properties are summarized in Section 3 for the reader's convenience. Such a notion of degree has been introduced in [8] (see also [7,9,10] for additional details).…”

Orientation of Fredholm maps

“…[4]). If f : V → X is orientable on an open neighborhood V of K we can define a BF-orientation on V as follows: choose a base point x α in each connected component V α of V and choose a finite rank operator D α such that Df (x α )+D α ∈ GL(X).…”

“…A different version of this degree, based on the orientation double covering of Φ 0 (X), was developed by Benivieri, Furi, Pera and others in a series of papers [2][3][4][5]. A precise description is given in Section 6.…”

“…From this point of view, the parity appears as the minimal geometric information needed for the construction of a degree while the various methods of orientation according to the preference of each author can be seen as an extra structure. This also explains why in approaches to orientation which do not use parity from the beginning it appears shortly later in calculations, maybe under a different name, e.g., "crossing numbers" in [38], which uses trivializations of the determinant line bundle or "sign jump" in [4] which uses orientation coverings.…”

Counting Zeros of C 1 Fredholm Maps of Index 1

“…Although the following result is implicit in [4], no proof of it is given there. For the sake of completeness, we prove it here.…”