Twisted monodromy homomorphisms and Massey products

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 \(\phi_*\).