By Vladimir Turaev (auth.)

Three-dimensional topology comprises colossal domain names: the examine of geometric constructions on 3-manifolds and the examine of topological invariants of 3-manifolds, knots, and so on. This publication belongs to the second one area. we will learn an invariant known as the maximal abelian torsion and denoted T. it truly is outlined for a compact delicate (or piecewise-linear) manifold of any size and, extra ordinarily, for an arbitrary finite CW-complex X. The torsion T(X) is a component of a undeniable extension of the crowd ring Z[Hl(X)]. The torsion T might be obviously thought of within the framework of straightforward homotopy thought. specifically, it's invariant below easy homotopy equivalences and will distinguish homotopy similar yet non­ homeomorphic CW-spaces and manifolds, for example, lens areas. The torsion T can be utilized additionally to tell apart orientations and so-called Euler constructions. Our curiosity within the torsion T is because of a specific position which it performs in third-dimensional topology. firstly, it's in detail with regards to a couple of basic topological invariants of 3-manifolds. The torsion T(M) of a closed orientated 3-manifold M dominates (determines) the 1st hassle-free excellent of 7fl (M) and the Alexander polynomial of 7fl (M). The torsion T(M) is heavily regarding the cohomology jewelry of M with coefficients in Z and ZjrZ (r ;::: 2). it's also regarding the linking shape on Tors hello (M), to the Massey items within the cohomology of M, and to the Thurston norm on H2(M).

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C), the matrix a(modI2) is obtained from 8 by replacing each entry of hk by hk - 1(mod12).

5. Lemma. Let 7r be a finitely generated group. Set H = HI (7r), G = H/TorsH and let pr : Z[H] - t Z[G] be the natural projection. IfrkH 22, then pr(E(7r)) C ~(7r) I(G). This lemma will be instrumental in Chapter IV. It improves the inclusion pr(E(7r)) C ~(7r) Z[G] which holds by the very definition of ~(7r). Proof. Present 7r by generators and relations (Xl, ... , Xm : rl, r2, ... ) with finite m 2 1 and at least m relations. Let A be the Alexander-Fox matrix of this presentation. We need to show that for any minor determinant D of A of order m - 1, pr(D) E ~(7r) I(G).

Weakly symmetric) is preserved under the transformations Lg dgg f---t Lg dga( 'IjJ(g) )g. 4. Remark. e) is yet to be proven, we indicate here another proof of the weak symmetry of the Alexander polynomials based on the classical duality for the Reidemeister-Franz torsions due to Franz and Milnor. Recall first their theorem. Let M be a compact orient able m-dimensional manifold. Set H = H1(M). Consider a field F and a ring homomorphism 'P : Z[H] ----t F equivariant with respect to the conjugation in Z[H] and an involution f f---t in F.

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