Torsions of 3-dimensional Manifolds
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).
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abelian Alexander polynomial algebraically Applying Assume Axiom bases basis boundary called canonical cells chain complex Chapter charge Choose circle claim clear Clearly components compute connected oriented 3-manifold Consider consists core Corollary corresponding defined definition denoted depend determined directed disc dual e e Eul(M element equal equivalent Eul(M Euler structures extends exterior finite follows formula framing function fundamental gives gluing Hence homology class homology orientation homomorphism ideal implies induced integer invariant Lemma matrix meridian norm Note obtain pair Pick projection proof proof of Theorem prove Q-homology sphere Recall relations Remark represented respectively restriction ring ring homomorphism Section sequence shows side solid tori splitting square surface surgery symmetric Theorem Tors torsion torsion function torus unique vector field yields
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Page 196 - Santa Barbara, USA Lectures on Algebraic Quantum Groups 2002. 360 pages. Softcover ISBN 3-7643-671 4-8 Audin, M., Universite Louis Pasteur et CNRS, Strasbourg, France / Cannas da Silva, A., Lisboa, Portugal / Lerman, E., University of Illinois at UrbanaChampaign, USA Symplectic Geometry of Integrable Hamiltonian Systems 2003.
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