Twisted monodromy homomorphisms and Massey products

Abstract

Let $ϕ: M\to M$ be a diffeomorphism of a $C^\infty$ compact connected manifold, and $X$ its mapping torus. There is a natural fibration $p:X\to S^1$, denote by $ξ\in H^1(X, \mathbb{Z})$ the corresponding cohomology class. Let $ρ:π_1(X)\to GL(n,\mathbb{C})$ be a representation, denote by $H^*(X,ρ)$ the corresponding twisted cohomology of $X$. Denote by $ρ_0$ the restriction of $ρ$ to $π_1(M)$, and by $ρ^*_0$ the antirepresentation conjugate to $ρ_0$. We construct from these data an automorphism of the group $H_*(M,ρ^*_0)$, that we call the twisted monodromy homomorphism $ϕ_*$. The aim of the present work is to establish a relation between Massey products in $H^*(X,ρ)$ and Jordan blocks of $ϕ_*$. Given a non-zero complex number $λ$ define a representation $ρ_λ:π_1(X)\to GL(n,\mathbb{C})$ as follows: $ρ_λ(g)=λ^{ξ(g)}\cdotρ(g)$. 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 $J_k(ϕ_*, λ)$ is equal to the maximal length of a non-zero Massey product of the form $\langle ξ, \ldots , ξ, x\rangle$ where $x\in H^k(X,ρ)$ (here the length means the number of entries of $ξ$). 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 $ϕ_*$.