Massey products in mapping tori

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 $\lambda\in \mathbb{Z}^*$. Consider the endomorphism $\phi_k^*$ induced by $\phi$ in the cohomology of $M$ of degree $k$, and denote by $J_k(\lambda)$ the maximal size of its Jordan block of eigenvalue $\lambda$. Define a representation $\rho_\lambda : \pi_1(X)\to\mathbb{C}^*$ by $\rho_\lambda (g) = \lambda^{p_*(g)}.$ Let $H^*(X,\rho_\lambda)$ be the corresponding twisted cohomology of $X$. We prove that $J_k(\lambda)$ is equal to the maximal length of a non-zero Massey product of the form $\langle \xi, \ldots , \xi, a\rangle$ where $a\in H^k(X,\rho_\lambda)$ (here the length means the number of entries of $\xi$). In particular, if $X$ is a strongly formal space (e.g. a K\"ahler manifold) then all the Jordan blocks of $\phi_k^*$ are of size 1. If $X$ is a formal space, then all the Jordan blocks of eigenvalue 1 are of size 1. This leads to a simple construction of formal but not strongly formal mapping tori. The proof of the main theorem is based on the fact that the Massey products of the above form 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 $\phi^*$.