Detecting Invertibility from the Topology of the Pre-images of Hyperplanes

In this work, we are concerned with finding topological conditions ensuring that a local diffeomorphism is bijective. A classical result in this direction is the well-known Hadamard-Plastock Theorem. It states that a Banach space local diffeomorphism f : X → X is bijective provided inf x∈X kDf(x)−1k−1 > 0. Although the proof of the Hadamard-Plastock theorem involves some technical details, it follows essentially from simple arguments involving covering spaces.

In recent years new topological and geometric ideas have been introduced in the subject of global invertibility, pushing the field in different directions. The emerging picture reveals that global invertibility is also influenced by more subtle topological phenomena. In particular, the work of Nollet and Xavier provided a substantial improvement to the Hadamard-Plastock theorem when dimX < ∞. Using degree theory, they showed that a local diffeomorphism f : Rn → Rn is bijective if there exists a complete Riemannian metric g on Rn such that for all unit vector v, inf x∈Rn kDf(x)∗vkg > 0. A short computation shows that this analytic condition implies the one in the Hadamard-Plastock theorem.

Arguments from elementary Morse theory show that under the conditions of the Nollet-Xavier theorem, the pre-images of affine hyperplanes H satisfy f−1(HR ∼= Rn. In particular, it follows that f−1(H) is acyclic, that is, f−1(H) has the homology of a point. The aim of this dissertation is to show that knowledge of the topology of the pre-images of hyperplanes alone is enough to detect global invertibility.

Theorem. A local diffeomorphism f : Rn → Rn is bijective if and only if the pre-image of every affine hyperplane is non-empty and acyclic.

Other results of similar nature are also established in this dissertation. The proof of our main theorem is based on some geometric constructions involving foliations and computation of intersection numbers between certain chain complexes. Our result also allows for an analytic corollary that is stronger than the Nollet-Xavier theorem in the sense that one can choose the complete metric g to suit the unit vector v.