Applications of minimal surfaces to the topology of threemanifolds william h. Introduction to the geometry and topology of manifolds i. He was among many other things a cartographer and many terms in modern di erential geometry chart, atlas, map, coordinate system, geodesic, etc. We will follow the textbook riemannian geometry by do carmo.
This book is an introduction to manifolds at the beginning graduate level. Let gen,n be the oriented grassmann manifold formed by all oriented n dimensional subspaces of rn. Incompressible surfaces in the figureeight knot complement. Proceedings of symposia in pure mathematics, issn 00820717. This theory generalizes thurstons theory of surface automorphisms, and reveals an intimate connection between dynamics, geometry and topology in 3 dimensions. Hence, in the computation of the norm of a homology class in m, it su. Introduction topology of 3manifolds and related topics. You have to spend a lot of time on basics about manifolds, tensors, etc.
Every oriented threemanifold can be obtained by this construction lickorish. Thurstons threedimensional geometry and topology, volume 1 princeton university press, 1997 is a considerable expansion of the first few chapters of these notes. Collapsing three manifolds under a lower curvature bound shioya, takashi and yamaguchi, takao, journal of differential geometry, 2000 examples of transversally complex submanifolds of the associative grassmann manifold enoyoshi, kanako and tsukada, kazumi, tsukuba journal of mathematics, 2019. Chapter 1 geometry and three manifolds with front page, introduction, and table of contents, ivii, 17 pdf ps ps. The main goal is to describe thurstons geometrisation of three manifolds, proved by perelman in 2002. In the study of surfaces it is helpful to take a geometric point of view. Progress in lowdimensional topology has been very quick in the last three decades, leading to the solutions of many difficult problems. In the early days of geometry nobody worried about the natural context in which the methods of calculus feel at home.
Outline overview milestones future directions outline of the talk i overview of geometry and topology of 3 manifolds i fields impacted by and impacting 3manifold topology i milestones in 3dimensional geometric topology. There are two topological processes to join 3 manifolds to get a new one. The geometry and topology of threemanifolds is a set of widely circulated but unpublished notes by william thurston from 1978 to 1980 describing his work on 3manifolds. Introduction to geometric group theory and 3manifold topology. There is a wellknown theorem in topology which deals with similar situations. The geometry and topology of three manifolds by william paul thurston. The geometry and topology of 3manifolds and gravity. In this note, we study the topology and the differential geometry of fivedimensional riemannian manifolds carrying such an so3ir structure, i. The goal of the course is to study the interplay between geometry, algebra and topology which occurs in geometric group theory. Geometric topology this area of mathematics is about the assignment of geometric structures to topological spaces, so that they look like geometric spaces. Akiyama, applications of nonstandard analysis to stochastic flows and heat kernels on manifolds. Numbers on the right margin correspond to the original editions page numbers. The geometry and topology of three manifolds is a set of widely circulated but unpublished notes by william thurston from 1978 to 1980 describing his work on 3 manifolds. Important types of 3manifolds are hakenmanifolds, seifertmanifolds, 3dimensional lens spaces, torusbundles and torus semibundles.
There are two topological processes to join 3manifolds to get a new one. The purpose of this book is to give an exposition of the socalled pseudo anosovtheory offoliations of 3manifolds. The main goal is to describe thurstons geometrisation of threemanifolds, proved by perelman in 2002. Lecture notes from princeton university 197880 on free shipping on qualified orders. In the the late 1970s thurston1 proposed a geometric classification of the topologies of closed three dimensional manifolds. Introduction to geometric group theory and 3manifold topology jean raimbault abstract. The sphere inherits a riemannian metric of 0 curvature in the complement of these 4 points, and. This theorygeneralizesthurstons theory of surface automorphisms, and reveals an intimate connection between dynamics, geometry and topology in 3 dimensions. Lecture notes geometry of manifolds mathematics mit. Future directions in 3manifold geometry and topology. Three dimensional manifolds, kleinian groups and hyperbolic geometry. Physically, one may imagine a mirror placed on the y. Until a few decades ago, a standard undergraduate course in topology consisted of a rigorous development of point set topology that was intended only for advanced mathematics majors headed for graduate school.
Lecture 1 notes on geometry of manifolds lecture 1 thu. Important types of 3 manifolds are haken manifolds, seifert manifolds, 3dimensional lens spaces, torusbundles and torus semibundles. Academic honesty is expected of all students in all examinations, papers, laboratory work, academic transactions and records. Riemannian, symplectic and poisson manifolds, lie groups, lie groupoids, lie algebroids and lierinehart algebras, poisson algebras. These notes, originally written in the 1980s, were intended as the beginning of a book on 3 manifolds, but unfortunately that project has not progressed very far since then. Watanabe, hamiltonian structure and formal complete integrability of thirdorder evolution equations of not normal type. The name or names attached to each question is that of the proposer, though many of. Introduction to topological manifolds springerlink.
The possible sanctions include, but are not limited to, appropriate grade penalties, course failure indicated on the transcript as a grade of e, course failure due to academic dishonesty indicated on the transcript as a grade of xe, loss of. The geometry and topology of threemanifolds wikipedia. By a classical result of eliashberg, contact 3 manifolds come in two flavors, flexible overtwisted and rigid tight. Applications of minimal surfaces to the topology of three. This book provides a selfcontained introduction to the topology and geometry of surfaces and three manifolds. The main reference will be algebraic topology by allen hatcher chapters 0, 1 and appendix, available here. Tejas kalelkar 1 introduction in this project i started with studying the classi cation of surface and then i started studying some preliminary topics in 3 dimensional manifolds.
Applications of minimal surfaces to the topology of three manifolds william h. Pims symposium on the geometry and topology of manifolds 29 june july 10, 2015 earth sciences building university of british columbia this conference will gather mathematicians working on a broad range of topics in the geometry and topology of manifolds and provide an opportunity for researchers and graduate students to learn about new. Thurston, on the geometry and dynamics of diffeomorphisms of surfaces, i. Notes on basic 3manifold topology cornell university. Outline overview milestones future directions outline of the talk i overview of geometry and topology of 3manifolds i fields impacted by and impacting 3manifold topology i milestones in 3dimensional geometric topology.
The notes introduced several new ideas into geometric topology, including orbifolds, pleated manifolds, and train tracks. Topology and geometry of threedimensional manifolds. However the reader should bear in mind that these pages are really just an early draft of the initial chapters of a real book on 3manifolds, which i had originally hoped to write. The topology of 3manifolds, heegaard distance and the mapping class group of a 2manifold. In may 2015, a conference entitled groups, geometry, and 3manifolds was held at the university of california, berkeley. In 3, chen showed that gen,n can be imbedded in the unit sphere of wedge product space. Foliations and the geometry of 3 manifolds by danny calegari oxford university press the book gives an exposition of the pseudoanosov theory of foliations of 3 manifolds. Yoshioka, the quasiclassical calculation of eigenvalues for the bochnerlaplacian on a line bundle. Geometry and topology of 3manifolds workshop list of participants alphabetically chris atkinson university of minnesota, morris jose ayala university of melbourne hyungryul baik kaist michael brandenbursky ben gurion university martin bridgeman boston college mark brittenham university of nebraskalincoln. The topology of 3manifolds, heegaard distance and the. Consider the nonstandard embedding of so3 into so5 given by the fivedimensional irreducible representation of so3, henceforth called so3ir. It should provide the reader with a better understanding of the physical properties of euclidean 3spacethe space in which we presume we live. Among the earlier highlights of this period was cassons. Gz zip tgz chapter 2 elliptic and hyperbolic geometry, 926 pdf ps ps.
This is the path we want to follow in the present book. So it seemed worthwhile to make this available electronically. In 1910 dehnlo published a proof showing, for a jordan curve on the boundary of a compact threedim. Pdf file of the 2007 version this is the current version. Survey articles by legendary mathematicians such as r. We will be particularly interested in the applications of these ideas to. In may 2015, a conference entitled groups, geometry, and 3 manifolds was held at the university of california, berkeley. Topology and geometry of 2 and 3 dimensional manifolds chris john may 3, 2016 supervised by dr.
Thurstons threedimensional geometry and topology, vol. In the case when xis not in the interior of the base triangle. The organizers asked participants to suggest problems and open questions, related in some way to the subject of the conference. Because of this relation, many questions which seem utterly hopeless from a purely topological point of view can be fruitfully studied. It is directed toward mathematicians interested in geometry who have had at least a beginning course in topology.
Find materials for this course in the pages linked along the left. Foliations and the geometry of 3manifolds by danny calegari oxford university press the book gives an exposition of the pseudoanosov theory of foliations of 3manifolds. Thurstons three dimensional geometry and topology, volume 1 princeton university press, 1997 is a considerable expansion of the first few chapters of these notes. It contains the essential topological ideas that are needed for the further study of manifolds, particularly in the context of differential geometry, algebraic topology, and related fields. A little more precisely it is a space together with a way of identifying it locally with a euclidean space which is compatible on overlaps.
Topology and geometry of threedimensional manifolds stephan tillmann version 8. In geometry and topology, all manifolds are topological manifolds, possibly with additional structure, such as a differentiable structure. There was no need to address this aspect since for the particular problems studied this was a nonissue. A manifold can be constructed by giving a collection of coordinate charts, that is a covering by open sets with homeomorphisms to a euclidean space, and patching functions. Gz zip tgz chapter 3 geometric structures on manifolds, 2743 pdf ps ps. In this chapter we will study geometry in the classical sense. Except for pagination, this version is identical with the published version we have had a longstanding interest in the way that structure in the mapping class group of a. Methods and applications part 2 the geometry and topology of manifolds graduate texts in mathematics 93. Collapsing threemanifolds under a lower curvature bound shioya, takashi and yamaguchi, takao, journal of differential geometry, 2000 examples of transversally complex submanifolds of the associative grassmann manifold enoyoshi, kanako and tsukada, kazumi, tsukuba journal of mathematics, 2019. In this paper, i will mention some applications of minimal surfaces to the geometry and topology of threemanifolds that i discussed in my lecture at the current developments in mathematics conference for 2004. These have been collected here, roughly divided by topic. On threemanifolds with bounded geometry 47 proposition 1. African institute for mathematical sciences south africa 272,296 views 27.
The first is the connected sum of two manifolds and. The completion of hyperbolic threemanifolds obtained from ideal polyhedra. Contact structures in three dimensions play an important role in topology of 3 and 4 manifolds. Threemanifolds may seem harder to understand at first. For instance, compact two dimensional surfaces can have a local geometry based on the sphere the sphere itself, and the projective plane, based on the euclidean plane the torus and the.
The scene as viewed by a person in this halfspace is like all of r3, with scenery invariant by the z 2 symmetry. Topology and geometry of 2 and 3 dimensional manifolds. The geometry and topology of threemanifolds download link. In this paper, i will mention some applications of minimal surfaces to the geometry and topology of three manifolds that i discussed in my lecture at the current developments in mathematics conference for 2004.68 253 1002 621 1367 1503 377 1369 992 1363 393 1351 1228 1270 306 278 373 1414 1465 255 1185 933 22 117 450 299 105 293 944 570 1286 1042 1146 1217 1405 1442 193 208 1041 748 341