Twisted monodromy homomorphisms and Massey products

Abstract

Let φ : M Ñ M be a diffeomorphism of a C 8 compact connected manifold, and X its mapping torus. There is a natural fibration p : X Ñ S 1 , denote by ξ P H 1 pX, Zq the corresponding cohomology class. Let ρ : π1pX, x0q Ñ GLpn, Cq be a representation (here x0 P M ); denote by H˚pX, ρq the corresponding twisted cohomology of X. Denote by ρ0 the restriction of ρ to π1pM, x0q, and by ρ0 the antirepresentation conjugate to ρ0. We construct from these data the twisted monodromy homomorphism φ˚of the group H˚pM, ρ0 q. This homomorphism is a generalization of the homomorphism induced by φ in the ordinary homology of M . The aim of the present work is to establish a relation between Massey products in H˚pX, ρq and Jordan blocks of φ˚.We have a natural pairing H˚pX, Cq b H˚pX, ρq Ñ H˚pX, ρq; one can define Massey products of the form xξ, . . . , ξ, xy, where x P H˚pX, ρq. The Massey product containing r terms ξ will be denoted by xξ, xyr; we say that the length of this product is equal to r. Denote by M k pρq the maximal length of a non-zero Massey product xξ, xyr for x P H k pX, ρq. Given a non-zero complex number λ define a representation ρ λ : π1pX, x0q Ñ GLpn, Cq as follows: ρ λ pgq " λ ξpgq¨ρ pgq. Denote by J k pφ˚, λq 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 M k pρ λ q " J k pφ˚, λq. In particular, φi s 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˚.