Massey products in mapping tori

TL;DR

This paper establishes a connection between Massey products in the cohomology of mapping tori and the Jordan normal form of the induced map, revealing conditions for formality and strong formality in such spaces.

Contribution

It provides a novel characterization of Jordan block sizes via Massey products in the cohomology of mapping tori, linking algebraic and topological properties.

Findings

Jordan block size equals maximal length of certain Massey products.

Strongly formal spaces have all Jordan blocks of size 1.

Constructs examples of formal but not strongly formal 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 $\phi^*$.