The theory of 3-manifolds has been revolutionised in the last few years by work of Thurston [66-70]. He has shown that geometry has an important role to play in the theory in addition to the use of purely topological methods. The basic aim of this article is to discuss the various geometries which arise and explain their significance for the theory of 3-manifolds. The idea is that many 3-manifolds admit 'nice' metrics which give one new insight into properties of the manifolds. For the purposes of this article, the nicest metrics are those of constant curvature. An observer in a manifold with a constant curvature metric will see the same picture wherever he stands and in whichever direction he looks. Such manifolds have special topological properties. However, we will also need to consider nice metrics which are not of constant curvature. In this article, I will explain what is meant by a 'nice' metric and describe their classification in dimension three which is due to Thurston. Then I will discuss some of the 3-manifolds which admit these nice metrics and the relationship between their geometric and topological properties. In this introduction all manifolds and metrics will be assumed to be smooth so that the objects of interest are all Riemannian manifolds. It has been known since the nineteenth century that in dimension two there is a very close relationship between geometry and topology. I will start by describing some basic facts about closed surfaces. I will discuss these in more detail in §1. Each

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The geometries of 3-manifolds

0. Introduction. An n-dimensional convex polytope is simple if the number of codimension-one faces meeting at each vertex is n. In this paper we investigate certain group actions on manifolds, which have a simple convex polytope as orbit space. Let P\" denote such a simple polytope. We have two situations in mind. (1) The group is Z, M\" is n-dimensional and (2) The group is T\", M is 2n-dimensional and M2\"/T\" P\". Up to an automorphism of the group, the action is required to be locally isomorphic to the standard representation of Z. on in the second case. In the first case, we call M a \"small cover\" of P\"; in the second, it is a \"toric manifold\" over P. First examples are provided by the natural actions of Z. and T\" on RP\" and CP\", respectively. In both cases the orbit space is an n-simplex. Associated to a small cover of P\", there is a homomorphism 2\" Z’ Z, where m is the number of codimension-one faces of P\". The homomorphism 2 specifies an isotropy subgroup for each codimension-one face. We call it a \"characteristic function\" of the small cover. Similarly, the characteristic function ofa toric manifold over pn is a map Z --) 7/\". A basic result is that small covers and toric manifolds over P\" are classified by their characteristic functions (see Propositions 1.7 and 1.8). The algebraic topology of these manifolds is very beautiful. The calculation of their homology and cohomology groups is closely related to some well-known constructions in commutative algebra and the combinatorial theory of convex polytopes. We discuss some of these constructions below. Let f denote the number of/-faces of P\" and let h denote the coefficient of \"in f(t 1). Then (fo, f,) is called the f-vector and (ho, h,) the h-vector of P\". The f-vector and the h-vector obviously determine one another. The Upper Bound Theorem, due to McMullen, asserts that the inequality h < (,-,{-x), holds for all n-dimensional convex polytopes with m faces of codimension one. In 1971 McMullen conjectured simple combinatorial conditions on a sequence (h0, h,) of integers necessary and sufficient for it to be the h-vector of a simple convex polytope. The sufficiency of these conditions was proved by Billera and Lee and necessity by Stanley (see [Bronsted] for more details and references). Research on

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Convex polytopes, Coxeter orbifolds and torus actions

Riemannian foliations

These notes are intended as an introduction to the theory of Coxeter groups. They closely follow my talk in the Lectures on Modern Mathematics Series at the Mathematical Sciences Center in Tsinghua University on May 10, 2013. They were prepared from the beamer presentation which I used during my talk. AMS classification numbers. Primary: 20F55, 20F65

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The geometry and topology of Coxeter groups

Introduction Polytopes Topology and combinatorics of simplicial complexes Commutative and homological algebra of simplicial complexes Cubical complexes Toric and quasitoric manifolds Moment-angle complexes Cohomology of moment-angle complexes and combinatorics of triangulated manifolds Cohomology rings of subspace arrangement complements Bibliography Index.

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Torus Actions and their Applications in Topology and Combinatorics

Seifert manifolds, plumbing, µ-invariant and orientation reversing maps

Smooth actions of the 2-dimensional torus group SO(2) x 50(2) on smooth, closed, orientable 4-manifolds are studied. A cross-sectioning theorem for actions without finite nontrivial isotropy groups and with either fixed points or orbits with isotropy group isomorphic to SO(2) yields an equivariant classification for these cases. This classification is made numerically specific in terms of orbit invariants. A topological classification is obtained for actions on simply connected 4-manifolds. It is shown that such a manifold is an equivariant connected sum of copies of complex projective space CP2, — CP2 (reversed orientation), S2 x S2 and the other oriented S2 bundle over S2. The latter is diffeomorphic (but not always equivariantly diffeomorphic) to CP2 # —CP2. The connected sum decomposition is not unique. Topological actions on topological manifolds are shown to reduce to the smooth case. In an appendix certain results are extended to torus actions on orientable 4dimensional cohomology manifolds. Introduction. In this article we begin the study of actions of the two-dimensional torus group 50(2) x SO(2) on 4-manifolds. Our chief motivation for this study is that it enables one to construct, in a systematic way, a large number of nicely behaved 4-manifolds. These 4-manifolds can be described in terms of certain weighted 2-manifolds (orbit spaces with their orbit structures). To write down all possible actions of the torus on 4-manifolds is not too difficult but it is entirely a different matter to characterize topologically the 4-manifold from the orbit data. We may think of this study as two separate problems. By the equivariant problem we mean being able to write down enough data (orbit space, orbit types, etc.) so that when two sets of data arising from two actions (not necessarily on the same manifold) are examined, one can decide whether or not there is a homeomorphism between the two manifolds which is equivariant with respect to both actions. This problem is considerably easier than the topological classification problem. By topological classification we mean the topological identification and classification of the manifolds arising in terms of their data used for deciding the first problem. Received by the editors December 15, 1969. AMS 1968 subject classifications. Primary 5560, 5747.

https://www.ams.org/tran/1970-152-02/S0002-9947-1970-0268911-3/S0002-9947-1970-0268911-3.pdf

Actions of the torus on 4-manifolds. I

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3-dimensional Lorentz space-forms and Seifert fiber spaces

(The manifold M6,g,h,t is an explicit sum of handles and p2 X S1's. The L'(Pt, v1)'s are lens spaces each with a specific, or "standard," action. This is described more precisely in the text.) Since each allowable unordered tuple corresponds uniquely to a distinct class of equivalent actions, we can, if we are able to " identify " the 3-manifold, determine all the inequivalent actions on a given 3-manifold. The corollary to Theorem 4 does just this for actions with fixed points. (For example, the manifold admitting the action (0; (o, 6, 3, 5); (9, 2), (7, 5), (8, 3), (15, 6)) admits exactly 5,632 strictly inequivalent actions with fixed points and no actions without fixed points.)

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Classification of the actions of the circle on 3-manifolds

Aspherical manifolds are closed manifolds which are A\"(w, l)'s. They play a significant role in many branches of mathematics. This paper constructs \"model\" aspherical manifolds for various classes of IT and investigates and surveys their groups of homeomorphisms. It is not known whether aspherical manifolds having the same fundamental groups as our model manifolds can topologically differ from them. If m contains a normal abelian subgroup then the model aspherical manifolds are special instances of injective Seifert fiber spaces. The group of (singular) fiber preserving homeomorphisms of Seifert fiber spaces are characterized and criteria obtained so that each self-homotopy equivalence may be deformed to such a homeomorphism. Many of our model aspherical manifolds satisfy these criteria. Other applications are given to group theory, differential geometry, complex manifolds as well as topology. We have also included a list of unsolved problems.

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Frank Raymond

Papers

Seifert manifolds, plumbing, µ-invariant and orientation reversing maps

Actions of the torus on 4-manifolds. I

3-dimensional Lorentz space-forms and Seifert fiber spaces

Classification of the actions of the circle on 3-manifolds

Deforming homotopy equivalences to homeomorphisms in aspherical manifolds