The Reidemeister Torsion of 3-Manifolds by Liviu I Nicolaescu

By Liviu I Nicolaescu

It is a cutting-edge advent to the paintings of Franz Reidemeister, Meng Taubes, Turaev, and the writer at the idea of torsion and its generalizations. Torsion is the oldest topological (but now not with recognize to homotopy) invariant that during its virtually 8 a long time of life has been on the middle of many very important and superb discoveries. up to now decade, within the paintings of Vladimir Turaev, new issues of view have emerged, which grew to become out to be the "right ones" so far as gauge concept is worried. The e-book gains normally the recent elements of this venerable suggestion. The theoretical foundations of this topic are provided in a mode available to these, who desire to research and comprehend the most principles of the idea. specific emphasis is upon the numerous and quite assorted concrete examples and methods which trap the subleties of the

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2 → G φˆ : G This induces by pullback a morphism φˆ ∗ : N(G˜1 ) → N(G˜2 ). 39 we deduce that the following diagram is commutative N(G1 ) F ˜ 1) w N(G φˆ ∗ φ u N(G2 ) u ˜ w N(G2 ). F 3. rank(G) ≥ 2. Set r := rank(G), H = Tors(G), F := G/H . Then Fˆ is an ˆ r-dimensional torus which can be identified with the identity component of G. ˆ := F(N(G)) ⊂ C(G, ˆ C). More In this case N(G) = Z[G], and thus N(G) ˆ coincides with the subring generated by the Fourier transforms of the precisely, N(G) Dirac functions δg .

1. 1. The CW-structure of a 2-torus. Observe that Xˆ ∼ = R2 and H = Z2 with (multiplicative) generators t1 and t2 . We choose the bases ci as follows. • c0 = αˆ = 0 ∈ R2 . • c1 = βˆ1 = I × {0} ⊂ R2 , βˆ2 = {0} × I ⊂ R2 , I = [0, 1] . • c2 = γˆ = I × I ⊂ R2 . 1 we deduce ∂ αˆ = 0, ∂ βˆ2 = (t2 − 1)α, ˆ ∂ γˆ = (1 − t2 )βˆ1 − (1 − t1 )βˆ2 . ˆ ∂ βˆ1 = (t1 − 1)α, Now choose b2 = {γˆ }, b1 = {βˆ1 }, Then [b2 /c2 ] = 1, [(∂b2 )b1 /c1 ] = det b0 = ∅ (1 − t2 ) 1 −(1 − t1 ) 0 [(∂b1 )b0 /c0 ] = (t1 − 1). We conclude that TT 2 ∼ 1 ∼ ±t1n1 t2n2 .

34. The following diagram is commutative. N(G0 ) F ˆ ∗0 , C) w C(G φ u N(G1 ) F u φˆ ∗ ˆ ∗1 , C). 6 Abelian harmonic analysis 33 Proof. The morphism φ is the restriction of the integration-along-fibers map φ∗ : C[G0 ] → C[G1 ] to the augmentation ideal, ker augG0 . Since augG1 (φ∗ (f )) = augG0 (f ), ∀f ∈ C[G0 ] we deduce that φ∗ (ker augG0 ) ⊂ ker augG1 . The proposition follows from the more general statement φˆ ∗ ◦ F = F ◦ φ∗ . ˆ 1 and f ∈ C[G0 ] we have Indeed for every χ ∈ G ˆ )) = f, φ(χ ˆ ) = φˆ ∗ (fˆ)(χ ) = fˆ(φ(χ ˆ = f (g)φ(χ)(g) g∈G0 f (g)) χ(g ¯ 1) = = g1 ∈G1 φ(g)=g1 f (g)χ (φ(g)) g∈G0 φ∗ (f )(g1 )χ¯ (g1 ) = F ◦ φ∗ (f ).

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