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Abstract Let ϕ : M → M be a diffeomorphism of a C ∞ compact connected manifold, and X its mapping torus. There is a natural fibration p : X → S 1 , denote by ξ ∈ H 1 (X , Z) the corresponding cohomology class. Let ρ : π 1 (X , x 0) → GL (n , C) be a representation (here x 0 ∈ M); denote by H ⁎ (X , ρ) the corresponding twisted cohomology of X. Denote by ρ 0 the restriction of ρ to π 1 (M , x 0) , and by ρ 0 ⁎ the antirepresentation conjugate to ρ 0. We construct from these data the twisted monodromy homomorphism ϕ ⁎ of the group H ⁎ (M , ρ ⁎ 0). This homomorphism is a generalization of the homomorphism induced by ϕ in the ordinary homology of M. The aim of the present work is to establish a relation between Massey products in H ⁎ (X , ρ) and Jordan blocks of ϕ ⁎. We have a natural pairing H ⁎ (X , C) ⊗ H ⁎ (X , ρ) → H ⁎ (X , ρ) ; one can define Massey products of the form 〈 ξ , ... , ξ , x 〉 , where x ∈ H ⁎ (X , ρ). The Massey product containing r terms ξ will be denoted by 〈 ξ , x 〉 r ; we say that the length of this product is equal to r. Denote by M k (ρ) the maximal length of a non-zero Massey product 〈 ξ , x 〉 r for x ∈ H k (X , ρ). Given a non-zero complex number λ define a representation ρ λ : π 1 (X , x 0) → GL (n , C) as follows: ρ λ (g) = λ ξ (g) ⋅ ρ (g). Denote by J k (ϕ ⁎ , λ) the maximal size of a Jordan block of eigenvalue λ of the automorphism ϕ ⁎ in the homology of degree k. The main result of the paper says that M k (ρ λ) = J k (ϕ ⁎ , λ). In particular, ϕ ⁎ is diagonalizable, if a suitable formality condition holds for the manifold X. This is the case if X a compact Kähler manifold and ρ is a semisimple representation. The proof of the main theorem is based on the fact that the above Massey products can be identified with differentials in a Massey spectral sequence, which in turn can be explicitly computed in terms of the Jordan normal form of ϕ ⁎. [ABSTRACT FROM AUTHOR]

Using the R-matrix formulation of the sl3 invariant of links, we compute the coloured sl3 generalised Jones polynomial for the trefoil. From this, the PSU(3) invariant of the Poincare´ homology sphere is obtained. This takes complex number values at roots of unity. The result obtained is formally an infinite sum, independent of the order of the root of unity, which at roots of unity reduces to a finite sum. This form enables the derivation of the PSU(3) analogue of the Ohtsuki series for the Poincare´ homology sphere, which it was shown by Thang Le could be extracted from the PSU(N) invariants of any rational homology sphere. [Copyright &y& Elsevier]