| Type of Document |
Dissertation |
| Author |
Balreira, Eduardo Cabral
|
| URN |
etd-04192006-093104 |
| Title |
Detecting Invertibility from the Topology of the Pre-images of Hyperplanes |
| Degree |
Doctor of Philosophy |
| Department |
Mathematics |
| Advisory Committee |
| Advisor Name |
Title |
| Frederico Xavier |
Committee Chair |
| Alan Seabaugh |
Committee Member |
| Brian Smyth |
Committee Member |
| Bruce Williams |
Committee Member |
| Liviu Nicolaescu |
Committee Member |
|
| Keywords |
- Global Analysis
- Geometric Topology
- Intersection Theory
- Invertibility
- Nonlinear Analysis
|
| Date of Defense |
2006-04-11 |
| Availability |
unrestricted |
Abstract
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 $f:X o X$ is bijective provided $ds inf_{xin X} |Df(x)^{-1}|^{-1}> 0$. 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 $dim X0$. 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~$finv(H)
obreak imes
obreakR
obreak
cong
obreakRn$. In particular, it follows that $finv(H)$ is extit{acyclic}, that is, $finv(H)$ has the homology of a point.
extit{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}.
oi extbf{Theorem. } extit{A local diffeomorphism $f:mathbb{R}^n omathbb{R}^n$ 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 $v$.
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