| Torsions of 3-Dimensional Manifolds 2002 Edition Contributor(s): Turaev, Vladimir (Author) |
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ISBN: 3034893981 ISBN-13: 9783034893985 Publisher: Birkhauser OUR PRICE: $52.24 Product Type: Paperback - Other Formats Published: October 2012 |
| Additional Information |
| BISAC Categories: - Mathematics | Mathematical Analysis - Medical - Mathematics | Geometry - General |
| Dewey: 514.3 |
| Series: Progress in Mathematics |
| Physical Information: 0.45" H x 6.14" W x 9.21" (0.67 lbs) 196 pages |
| Descriptions, Reviews, Etc. |
| Publisher Description: Three-dimensional topology includes two vast domains: the study of geometric structures on 3-manifolds and the study of topological invariants of 3-manifolds, knots, etc. This book belongs to the second domain. We shall study an invariant called the maximal abelian torsion and denoted T. It is defined for a compact smooth (or piecewise-linear) manifold of any dimension and, more generally, for an arbitrary finite CW-complex X. The torsion T(X) is an element of a certain extension of the group ring Z Hl(X)]. The torsion T can be naturally considered in the framework of simple homotopy theory. In particular, it is invariant under simple homotopy equivalences and can distinguish homotopy equivalent but non- homeomorphic CW-spaces and manifolds, for instance, lens spaces. The torsion T can be used also to distinguish orientations and so-called Euler structures. Our interest in the torsion T is due to a particular role which it plays in three-dimensional topology. First of all, it is intimately related to a number of fundamental topological invariants of 3-manifolds. The torsion T(M) of a closed oriented 3-manifold M dominates (determines) the first elementary ideal of 7fl (M) and the Alexander polynomial of 7fl (M). The torsion T(M) is closely related to the cohomology rings of M with coefficients in Z and ZjrZ (r; 2). It is also related to the linking form on Tors Hi (M), to the Massey products in the cohomology of M, and to the Thurston norm on H2(M). |