Twisted monodromy homomorphisms and Massey products

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→S1, denote by ξ∈H1(X,Z) the corresponding cohomology class. Let ρ:π1(X,x0)→GL(n,C) be a representation (here x0∈M); denote by H⁎(X,ρ) the corresponding twisted cohomology of X. Denote by ρ0 the restriction of ρ to π1(M,x0), and by ρ0⁎ the antirepresentation conjugate to ρ0. We construct from these data the twisted monodromy homomorphismϕ⁎ 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 Mk(ρ) the maximal length of a non-zero Massey product 〈ξ,x〉r for x∈Hk(X,ρ). Given a non-zero complex number λ define a representation ρλ:π1(X,x0)→GL(n,C) as follows: ρλ(g)=λξ(g)⋅ρ(g). Denote by Jk(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 Mk(ρλ)=Jk(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⁎.