The Invariance of Domain for C1 Fredholm maps of index zero

Is the one-to-one continuous image of an open set in R still open? This problem is the so called invariance of domain and, as H. Freudenthal pointed out in his essay on the history of topology [6] (see also [12]), it has been considered of a great importance at the beginning of this (20) century “for it was essential for the justification of the Poincaré continuation method in the theory of automorphic functions and the uniformization (of analytic functions).” As it is well-known, a positive answer to the problem has been given by Brouwer in 1912 ([3]). This result is usually obtained as a consequence of the Odd Mapping Theorem, as it is shown, for instance, in [5]. A simpler proof can be done by using the Jordan Separation Theorem (see, e.g., [7] or [10]). The extension of the Domain Invariance Theorem to Banach spaces is due to Schauder [9], who proved that if Ω is a bounded open subset of a Banach space E and f : Ω → E is a one-to-one map of the form f = I − h with I the identity of E and h : Ω → E compact, then f(Ω) is open. Again this result can be deduced by the infinite dimensional version of the Odd Mapping Theorem. A proof displaying this connection can be found, for instance, in [2], [5] or [7]. The result of Schauder is, clearly, a nonlinear counterpart of the wellknown Fredholm alternative.