3-manifolds (Google eBook)
A careful and systematic development of the theory of the topology of 3-manifolds, focusing on the critical role of the fundamental group in determining the topological structure of a 3-manifold ... self-contained ... one can learn the subject from it ... would be very appropriate as a text for an advanced graduate course or as a basis for a working seminar. --Mathematical Reviews For many years, John Hempel's book has been a standard text on the topology of 3-manifolds. Even though the field has grown tremendously, the book remains one of the best and most popular introductions to the subject. The theme of this book is the role of the fundamental group in determining the topology of a given 3-manifold. The essential ideas and techniques are covered in the first part of the book: Heegaard splittings, connected sums, the loop and sphere theorems, incompressible surfaces, free groups, and so on. Along the way, many useful and insightful results are proved, usually in full detail. Later chapters address more advanced topics, including Waldhausen's theorem on a class of 3-manifolds that is completely determined by its fundamental group. The book concludes with a list of problems that were unsolved at the time of publication. Hempel's book remains an ideal text to learn about the world of 3-manifolds. The prerequisites are few and are typical of a beginning graduate student. Exercises occur throughout the text.
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2-sphere 2-sphere bundle abelian annulus assume boundary components bounds a 2-cell bundle over S1 Choose compact 3-manifold completes the proof conclusion conjugate connected sum contradiction covering map covering translations cube with handles denote double cover double curve element epimorphism fake 3-cell ffj(F ffj(M fiber bundle finite index finite sheeted follows free group free product fundamental group genus Heegaard splitting hence Hj(M homeomorphism homology homotopy 3-sphere homotopy equivalent I-bundle incompressible surface induced infinite cyclic irreducible isomorphism isotopic LEMMA Let F loop theorem manifold map f Math monic nonorientable nonseparating nontrivial normal subgroup Note obtain P2-irreducible pairwise disjoint Poincare conjecture position map prime factorization projective plane properly embedded Q contains regular neighborhood residually finite S1 x S1 satisfying Seifert fibered space simple closed curve simplex solid torus sphere theorem subcomplex subgroup of finite submanifold Suppose surface F torsion free triangulation twisted I-bundle universal cover
Page 190 - JHC WHITEHEAD, A certain open manifold whose group is unity, Quart.