In this thesis we study the space L = L(M,N) of all Lagrangian mappings of a fixed closed surface or 3-manifold M to respectively another surface or 3-manifold N. In most cases we are assuming both M and N oriented. We are looking for local (order 1) invariants of such generic maps, that is, for those whose increments along generic paths in L are completely determined by diffeomorphism types of the local bifurcations of the caustics in N. Such invariants are dual to trivial cycles supported on the discriminantal hypersurface ? in L. The duality here is in the sense of the increment of an invariant along a generic path γ in L is the index of intersection of γ with the cycle, and the triviality means that if γ is a loop then its index of intersection with the cycle must vanish. For surfaces, we obtain a complete description of the spaces of discriminantal cycles, possibly non-trivial. For N = R² and the subset of maps in L without corank 2 singularities, this description implies that any rational local invariant itself is a linear combinations of the numbers of various singular points of the caustics and of the Ohmoto-Aicardi linking invariant of ordinary maps between surfaces. Using our discriminantal cycles, we also prove Ohmoto's conjecture about non-contractability of a certain loop in L(S²,R²). Our surface results are now published in [16]. For oriented 3-manifolds, we prove that the space of all rational local invariants is ten-dimensional and spanned by the numbers of various isolated-type singularities of the caustics and the Euler characteristic of the critical point set. We also show that the rank of the space of the mod2 invariants has dimension 16. The results of the thesis are based on our study of generic one- and two-parameter families of caustics. In our 3-dimensional constructions, we had to analyse generic projections of the D6 and E6 caustics to the plane. Nothing anyhow close to this rather delicate analysis has been done before, and it occupies nearly half the thesis.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:722036 |
| Date | January 2017 |
| Creators | Gallagher, Katy |
| Publisher | University of Liverpool |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | http://livrepository.liverpool.ac.uk/3006103/ |
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