The existence of slices in G-spaces, when G is a Lie group

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Title: The existence of slices in G-spaces, when G is a Lie group
Author: Karppinen, Sini
Contributor: Helsingin yliopisto, Matemaattis-luonnontieteellinen tiedekunta, Matematiikan ja tilastotieteen laitos
Thesis level:
Abstract: The theory of transformations groups is the theory of symmetries of a set. Formally, a symmetry of a set is a bijective map from the set to itself. If the set has some kind of mathematical structure, then we are naturally interested in those symmetries which preserve the given structure. For example, the symmetries of a topological space are homeomorphisms. When the transformations of a set form a group, we call the group a transformation group and the set a G-set. We also say that the group G acts on the set X. The study of these groups will reveal a lot about the set itself. In this thesis we are interested in topological transformation groups i.e. the transformation groups of topological spaces.We will, in particular, study the case where the topological space X is completely regular and where the transformation group G is a Lie group. The theory of Lie groups is vast and they have a well-understood structure. Our main goal is to present and prove the so-called slice theorem which is one of the most important results in the theory of transformation groups. A slice in a G-space X characterizes the action of G locally in an invariant neighbourhood of an orbit of X. The slice theorem i.e. the fact that there exists a slice at every point in a G-space X was first proved in the case where G is a compact Lie group. This was done in the 1950s by Gleason, Koszul, Montgomery and Yang and in full generality by Mostow. In order to prove the existence of slices in the case of non-compact Lie groups, the way that G acts needs to somehow be restricted. It turns out that proper action is the right way to do this. The existence of slices for proper actions of noncompact Lie groups was firsst proved by Palais in 1961. Abels and Lütkepohl presented a different kind of proof in 1977. We will present these two proofs in detail and compare them. We will present the machinery needed for the two proofs. The first four chapters are dedicated to the general theory of topological transformation groups, Lie groups and their representations and infinite-dimensional manifolds. Chapter five is dedicated to proper actions. Then in chapter six the two different proofs for the existence of slices for proper actions of non-compact Lie groups arepresented. After the detailed presentations of the proofs, we will compare them in a more general level. We will also present some applications of the slice theorem and consider possible ways to generalize the slice theorem for non-Lie groups.
Date: 2016-10-01
Subject: transformaatioryhmät, Lien-ryhmät, viipale
Discipline: Mathematics

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