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 group Alexander polynomial augmentation ideal Axiom basis bijective bilinear form boundary homomorphism C Q(H chain complex claim closed connected oriented cohomology compact connected orientable components compute connected oriented 3-manifold Consider consists of tori Corollary CW-complex CW-decomposition cyclic groups defined denoted determined directed solid tori e e Eul(M element equal Euler structures exterior family of cells finite fundamental family gluing group homomorphism group ring h e H Hf(M Hl(M homology class homology orientation homotopy implies inclusion homomorphism induced integer invariant isomorphism Laurent polynomial Lemma matrix meridian non-singular vector field non-zero oriented link oriented three-dimensional Pr(M proof of Theorem Q-homology sphere represented resp ring homomorphism S1 x S1 Section Seiberg-Witten Seiberg-Witten invariants solid tori spinc structure spinc(M square volume form surgery formula symmetric tangent Theorem 2.2 Thurston norm Tors H torsion function torus tubular neighborhood vect(M vector field
Page 194 - K. Walker [An extension of Casson's invariant, (Annals of Math. Studies 126, Princeton University Press, Princeton, NJ) (1992; Zbl 0752.57011)].