Twisted monodromy homomorphisms and Massey products

Abstract

Let $\phi: M\to M$ 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 $\xi\in H^1(X, \mathbb{Z})$ the corresponding cohomology class. Let $\rho:\pi_1(X)\to GL(n,\mathbb{C})$ be a representation, denote by $H^*(X,\rho)$ the corresponding twisted cohomology of $X$. Denote by $\rho_0$ the restriction of $\rho$ to $\pi_1(M)$, and by $\rho^*_0$ the antirepresentation conjugate to $\rho_0$. We construct from these data an automorphism of the group $H_*(M,\rho^*_0)$, that we call the twisted monodromy homomorphism $\phi_*$. The aim of the present work is to establish a relation between Massey products in $H^*(X,\rho)$ and Jordan blocks of $\phi_*$. Given a non-zero complex number $\lambda$ define a representation $\rho_\lambda:\pi_1(X)\to GL(n,\mathbb{C})$ as follows: $\rho_\lambda(g)=\lambda^{\xi(g)}\cdot\rho(g)$. Denote by $J_k(\phi_*, \lambda)$ the maximal size of a Jordan block of eigenvalue $\lambda$ of the automorphism $\phi_*$ in the homology of degree $k$. The main result of the paper says that $J_k(\phi_*, \lambda)$ is equal to the maximal length of a non-zero Massey product of the form $\langle \xi, \ldots , \xi, x\rangle$ where $x\in H^k(X,\rho)$ (here the length means the number of entries of $\xi$). In particular, $\phi_*$ is diagonalizable, if a suitable formality condition holds for the manifold $X$. This is the case if $X$ a compact K\"ahler manifold and $\rho$ 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 $\phi_*$.