Notes on Geometry and 3-Manifolds by Walter D. Neumann

Notes on Geometry and 3-Manifolds

Walter D. Neumann

Appendices by Paul Norbury

This is a preprint version. The final version was published, with minor changes, as pp. 191--267 in Papers from the Summer School held in Budapest, 1998 and EMS Summer School No. 1 on Algebraic Geometry held in Eger, 1996. Edited by Károly Böröczky, Jr., Walter Neumann and András Stipsicz. Bolyai Society Mathematical Studies, 8. János Bolyai Mathematical Society, Budapest, 1999, ISBN 963-8022-92-2.

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Distributed with permission from János Bolyai Mathematical Society.

Preface

These are a slightly revised version of the course notes that were distributed during the course on Geometry of 3-Manifolds at the Turan Workshop on Low Dimensional Topology in Budapest August 1998.

The lectures and tutorials did not discuss everything in these notes. The notes were intended to provide also a quick summary of background material as well as additional material for "bedtime reading". There are "exercises" scattered through the text, which are of very mixed diffculty. Some are questions that can be quickly answered. Some will need more thought and/or computation to complete. Paul Norbury also created problems for the tutorials, which are given in the appendices. There are thus many more problems than could be addressed during the course, and the expectation was that students would use them for self study and could ask about them also after the course was over.

For simplicity in this course we will only consider orientable 3-manifolds. This is not a serious restriction since any non-orientable manifold can be double covered by an orientable one.

In Chapter 1 we attempt to give a quick overview of many of the main concepts and ideas in the study of geometric structures on manifolds and orbifolds in dimension 2 and 3. We shall fill in some "classical background" in Chapter 2. In Chapter 3 we then concentrate on hyperbolic manifolds, particularly arithmetic aspects.

Lecture Plan

Geometric Structures
Proof of JSJ decomposition
Commensurability and Scissors congruence
Arithmetic invariants of hyperbolic 3-manifolds
Scissors congruence revisited: the Bloch group

Introduction

There are many ways of defining what is meant by a "geometric structure" on a manifold in the sense that we mean. We give a precise definition below. Intuitively, it is a structure that allows us to do geometry in our space and which is locally homogeneous, complete, and of finnite volume.

"Locally homogeneous" means that the space looks locally the same, where ever you are in it. I.e., if you can just see a limited distance, you cannot tell one place from another. On a macroscopic level we believe that our own universe is close to locally homogeneous, but on a smaller scale there are certainly features in its geometry that distinguish one place from another. Similarly. the surface of the earth, which on a large scale is a homogeneous surface (a 2-sphere) has on a smaller scale many little wrinkles and bumps, that we call valleys and mountains, that make it non-locally-homogeneous.

"Complete" means that you cannot fall off the edge of the space, as european sailors of the middle ages feared might be possible for the surface of the earth. We assume that our universe is complete, partly because anything else is pretty unthinkable.

The "finite volume" condition refers to the appropriate concept of volume. This is n-dimensional volume for an n- dimensional space, i.e., area when n = 2. Most cosmologists and physicists want to believe that our universe has finite volume.

Another way of thinking of a geometric structure on a manifold is as a space that is modeled on some "geometry". That is, it should look locally like the given geometry. A geometry is a space in which we can do geometry in the usual sense. That is, we should be able to talk about straight lines, angles, and so on. Most fundamental is that we be able to measure length of "reasonable" (e.g. smooth) curves. Then one can define a "straight line" or geodesic as a curve which is the shortest path between any two suffciently nearby points on the curve, and it is then not hard to define angles and volume and so on. We require a few more conditions of our geometry, the most important being that it is homogeneous, that is, it should look the same wherever you are in it. Formally, this means that the isometry group of the geometry, the group of length preserving invertible maps of the geometry to itself, should act transitively on the geometry. A consequence of this is that the geometry is complete -- you cannot fall off the edge. Another condition we require of a geometry is that it be simply-connected -- any closed loop should be continuously deformable to a point of the space; we will come back to this later.

We can describe a geometry by giving its underlying space and the element ds of arc-length. The length of a smooth curve is the path integral of ds along it.

Date received: September 19, 2000.