Orientation of Fredholm maps

Abstract

We introduce a notion of orientation of a Fredholm map on a given compact subset of its domain and show that various approaches to orientation have as outcome the same class of orientable maps.

Infinite-dimensional degree theory and stochastic analysis

“…The most appropriate approach theoretically is to define orientability of the map F itself. This has been done in several ways (see [5], [45], [58]). For us it will suffice to define an orientation of a JFPTA proper Φ 0 -map F : P → Q to be an assignment of a sign, sgn T x F in {−1, 1}, to every regular point x ∈ P in such a way that:…”

The family index theorem and bifurcation of solutions of nonlinear elliptic BVP

“…The groups J(S q ) are finite cyclic groups whose orders have been computed by Adams and others. In Theorem 1.4.1 of [17], under the assumption that the principal part of the boundary operator B λ (x, D) is independent of λ and that the principal part of the interior operator L λ (x, D) is independent of λ near the boundary, we computed the index of bifurcation obtaining in this way sufficient conditions for bifurcation in terms of a number defined as the integral of a differential form constructed explicitly from the principal symbol of the linearization L λ (x, D).…”

“…The purpose of the present paper is to extend Theorem 1.4.1 of [17] to the case in which also the principal symbol of the boundary operator is parameter dependent.…”

“…In [17] we introduced an index of bifurcation which counts algebraically the bifurcation points of a family of nonlinear Fredholm maps parametrized by open subsets of a compact manifold or polyhedron Λ. The index takes values in a finite abelian group J(Λ) associated to the parameter space.…”

“…In [17] we proved a parametrized version of the Agranovich reduction in order to recast the calculation of the index bundle of the family of linearizations along the trivial branch to that of a family of pseudo-differential operators on R n . Then we applied the Atiyah-Singer family index theorem to the latter.…”