Hempel, tao li, yair minsky, yoav moriah and richard weidmann. Introduction topology of 3manifolds and related topics. Let each face be identi ed with its opposite face by a translation without twisting. From the prime decomposition of closed 3manifolds m it follows that if. Our goal is to study compact, oriented 3manifolds and their splittings in terms of the geometry and combinatorics of cs. This article gives an account of those properties which have so far received sufficient attention, especially those concerning the diffeomorphism groups of 3 manifolds. The geometries of 3manifolds 403 modelled on any of these. Coverings of 3manifolds by open balls and two open solid tori j. Syllabus for introduction to hyperbolic 2 and 3manifolds. Find materials for this course in the pages linked along the left. The theme of this book is the role of the fundamental group in determining the topology of a given 3.
Heegaard splittings, connected sums, the loop and sphere theorems, incompressible surfaces, free groups, and so on. Later chapters address more advanced topics, including waldhausens theorem on a class of 3manifolds that is completely determined by its fundamental group. September 2009 page 1 of 3 product data sheets numerical index product no. Introduction to 3 manifolds 5 the 3 torus is a 3 manifold constructed from a cube in r3. The virtually special machine currently o ers no e ective method for determining, given an arbitrary hyperbolic 3 manifold m, the index of a subgroup of. We will be primarily interested in the case of closed 3manifolds v x and v y handlebodies. In 34 hyam rubinstein gives a personal collection of problems on 3manifolds. 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 3 manifolds, which i had originally hoped to write. Recall that a closed 3manifold m is geometric if it admits a riemannian metric such that the universal cover is homogeneous. In canonical quantum gravity certain topological properties of 3 manifolds are of interest. This book grew out of a graduate course on 3manifolds and is intended for a mathematically experienced audience that is new to lowdimensional topology. It is not clear whether distance survives to any sort of meaningful invariant for 3manifolds.
Thurston the geometry and topology of 3manifolds vii. Combinatorial problems and exercises laszlo lovasz. Primitive stable closed hyperbolic 3manifolds sciencedirect. Hyam rubinstein please note, due to essential maintenance online purchasing will not be possible between 03.
The book concludes with a list of problems that were unsolved at the time of publication. Next we give the sufficient conditions that fgip for 3manifold groups is preserved under torus sums or annulus sums and connect this result with a conjecture by hempel 4. Other related books on the mathematics of 3manifolds include 3manifolds by j. For compact 2 manifolds assume orientable, genus g 2, with nonempty boundary and free nonabelian fundamental group, there are lots of nitesheeted covering spaces that arise from group theory since f 2 surjects onto g, where gis nite and nonabelian. Jul 01, 2019 a pleasant feature of 3 manifolds, in contrast to higher dimensions, is that there j hempel. If a closed 3 manifold m can be covered by three open balls, then m is a connected sum of s3 and finitely many s2bundles over s1. Metric manifolds international winter school on gravity and light 2015 duration. Topology of general 3 manifolds see also 57mxx citation hempel, john. A 3 manifold can be thought of as a possible shape of the universe. Jul 18, 2019 a pleasant feature of 3 manifolds, in contrast to higher dimensions, is that there j hempel. Ja combinatorial interpretation of the third integral homology of a group. Topology of general 3manifolds see also 57mxx citation hempel, john. The reader is referred to 37 for a comprehensive survey of geometric 3 manifolds. For example2 x s, s1 has universal coverin2 xg u, s which is not homeomorphic t3 oor s u3.
Combinatorial squashings, 3manifolds, and the third. Let gbe the class of fundamental groups of closed geometric 3 manifolds. H 2 whose hempel distance is greater than k and which has rbounded combinatorics, the manifold m is hyperbolic and the representation. Here, and throughout these lectures, manifold will always mean a smooth, compact, connected, orientable manifold, we will not assume though that manifolds are closed. A note on hempelmcmillan coverings of 3manifolds request pdf. Princeton university press, university of tokyo press, 1976. Aug 12, 2019 a pleasant feature of 3 manifolds, in contrast to higher dimensions, is that there j hempel. The geometries of 3 manifolds 403 modelled on any of these. A clear presentation of seven of thurstons eight possible geometric structures on 3 manifolds, all but hyperbolic geometry, the most subtle case by far. Hempels book remains an ideal text to learn about the world of 3manifolds. For any 3manifold admitting a genusg heegaard splitting m h 1. It is a 3dimensional manifold because in a neighborhood of any given point it takes exactly three coordinates to specify a nearby point. We give a summary of these properties and list some old and new results concerning them.
The exposition begins with the definition of a manifold, explores possible additional structures on manifolds, discusses the classification of surfaces, introduces key foundational results for. Incompressible surfaces and the topology of 3dimensional manifolds volume 55 issue 1 iain r. This book grew out of a graduate course on 3 manifolds and is intended for a mathematically experienced audience that is new to lowdimensional topology. However adding a cancelling pair of handles reduces the distance of the splitting to zero. With the geometrization conjecture, this now holds for any compact and orientable 3manifold. In c we have a splitting into a gts 11,1 3, which is a solid torus, and an obl x 1, y 1, which by comment 6. Download our jeet tamari book pdf file ebooks for free and learn more about jeet tamari book pdf file. It is not clear whether distance survives to any sort of meaningful invariant for 3 manifolds. For many years, john hempel s book has been a standard text on the topology of 3 manifolds.
Coverings of 3manifolds by open balls and two open solid tori. Hempel and mcmillan hm proved that if m is covered by three open balls, then m is a. Hempel s book remains an ideal text to learn about the world of 3 manifolds. In mathematics, a 3 manifold is a space that locally looks like euclidean 3 dimensional space. If a closed 3manifold m can be covered by three open balls, then m is a connected sum of s3 and finitely many s2bundles over s1. In the other direction, 3manifolds have deep geometric structure, for example the.
Fundamental group, presentations, free differential calculus secondary. However2 x, u s an sd 2xsi each possesses a very natural metric which is simply the product of the standard metrics. Just as a sphere looks like a plane to a small enough observer, all 3 manifolds look like our universe does to a small enough observer. This in turn implies linear geodesic residual niteness growth for every closed hyperbolic 3 manifold, see 16, lemma 6. The topology of 3manifolds, heegaard distance and the.
Request pdf a note on hempelmcmillan coverings of 3manifolds motivated by the concept of acategory of a manifold introduced by clapp and puppe, we give a different proof of a slightly. Incompressible surfaces and the topology of 3dimensional. Syllabus for introduction to hyperbolic 2 and 3manifolds math 8790, spring 2014 albert marden february 4, 2014 some references. For many years, john hempels book has been a standard text on the topology of 3manifolds. The theme of this book is the role of the fundamental group in determining the topology of a given 3manifold. Let gbe the class of fundamental groups of closed geometric 3manifolds. A local signature for fibered 4manifolds with a finite group action sato, masatoshi, tohoku mathematical journal, 20. Hempel proved that haken manifolds have residually finite fundamental groups. So it seemed worthwhile to make this available electronically.
The theme of this book is the role of the fundamental group in determining the topology of a given 3 manifold. A topological space x is a 3manifold if it is a secondcountable hausdorff space and if every point in x has a neighbourhood that is homeomorphic to euclidean 3space mathematical theory of 3manifolds. With the geometrization conjecture, this now holds for any compact and orientable 3 manifold. Lecture notes geometry of manifolds mathematics mit.
Next we give the sufficient conditions that fgip for 3 manifold groups is preserved under torus sums or annulus sums and connect this result with a conjecture by hempel 4. We will make additional assumptions arising from the following reductions. Combinatorial squashings, 3manifolds, and the third homology. For compact 2manifolds assume orientable, genus g 2, with nonempty boundary and free nonabelian fundamental group, there are lots of nitesheeted covering spaces that arise from group theory since f 2 surjects onto g, where gis nite and nonabelian. Jun 21, 2019 hempels book remains an ideal text to learn about the world of 3manifolds. Motivated by the concept of acategory of a manifold introduced by clapp and puppe, we give a different proof of a slightly generalized theorem of hempel and mcmillan.
The topological, piecewiselinear, and smooth categories are all equivalent in three dimensions, so little distinction is made in whether we are dealing with. Later chapters address more advanced topics, including waldhausens theorem on a class of 3 manifolds that is completely determined by its fundamental group. Pdf file of the 2007 version this is the current version. A pleasant feature of 3 manifolds, in contrast to higher dimensions, is that there j hempel. The reader is referred to 37 for a comprehensive survey of geometric 3manifolds. Introduction to 3manifolds 5 the 3torus is a 3manifold constructed from a cube in r3. Coverings of3manifolds by opensubsets hempel and mcmillan showed that a closed 3manifold that can be covered by three open balls is a connected sum of s3ands2bundles over s1. Recall that a closed 3 manifold m is geometric if it admits a riemannian metric such that the universal cover is homogeneous. This article gives an account of those properties which have so far received sufficient attention, especially those concerning the diffeomorphism groups of 3manifolds.
You can imagine this as a direct extension from the 2torus we are comfortable with. Pdf coverings of 3manifolds by three open solid tori. Thurston, a simply connected 3 manifold is s3 if it is the sum of a solid torus and the complement of a torus knot, proc. We can assume that s is not a sphere or a disc, since otherwise theorem 1. A list of recommended books in topology cornell university. Hempel 1976, knots, links, braids and 3manifolds by prasolov and sosinskii 1997, algorithmic topology and classification of 3manifolds by s. Deforming threemanifolds with positive scalar curvature. M is free then m is a connected sum of s2bundles over s1 and a. Heegaard splittings of 3manifolds haifa 2005 problems. The geometry and topology of threemanifolds electronic version 1. Max dehn used this idea in 1910 to produce manifolds with the same homology as the 3sphere.
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