Massey products in mapping tori

Abstract

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 λ P C˚.Consider the endomorphism φk induced by φ in the cohomology of M of degree k, and denote by J k pλq the maximal size of its Jordan block of eigenvalue λ. Define a representation ρ λ : π1pXq Ñ C˚; ρ λ pgq " λ p˚pgq ; let H˚pX, ρ λ q be the corresponding twisted cohomology of X. We prove that J k pλq is equal to the maximal length of a non-zero Massey product of the form xξ, . . . , ξ, ay where a P H k pX, ρ λ q (here the length means the number of entries of ξ).In particular, if X is a strongly formal space (e.g. a Kähler manifold) then all the Jordan blocks of φk 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 φ˚.

Twisted monodromy homomorphisms and Massey products

“…Relations with other works. The case of the trivial representation ρ 0 was settled in the author's paper [14]. The diagonalizability of the monodromy homomorphism in the ordinary homology for Kähler manifolds was also proved by N. Budur, Y. Liu, B. Wang [2].…”

“…Remark 6.3. The particular case of the trivial representation χ corresponds to Theorems 3.1 and 5.1 of the paper [14]. 6.4.…”

On the generalization of Inoue manifolds

“…In Section 4.3 we show that if M is non-diagonalizable, then the manifold T M and its cartesian powers do not admit a structure of Kähler manifold. The proof is based on a theorem from [14] asserting that the monodromy of a fibration of a Kähler manifold over a circle is diagonalizable 1 .…”

“…, ξ, yy where y P H k (Y, C). It is proved in [14] that the maximal size of a Jordan block with eigenvalue 1 of the monodromy…”

“…The monodromy homomorphism of this fibration equals IdˆM J . This matrix is not diagonalizable,and the main theorem of [14] implies that (T M ) l does not admit a Kähler structure. □…”