Free abelian covers, short loops, stable length, and systolic inequalities

Abstract

We explore the geometry of the Abel–Jacobi map f from a closed, orientable Riemannian manifold X to its Jacobi torus $$H_1(X;{\mathbb{R}})/ H_1(X;{\mathbb{Z}})_{{\mathbb{R}}}$$ . Applying M. Gromov’s filling inequality to the typical fiber of f, we prove an interpolating inequality for two flavors of shortest length invariants of loops. The procedure works, provided the lift of the fiber is non-trivial in the homology of the maximal free abelian cover, $$\overline{X}$$ , classified by f. We show that the finite-dimensionality of the rational homology of $$\overline{X}$$ is a sufficient condition for the homological non-triviality of the fiber. When applied to nilmanifolds, our “fiberwise” inequality typically gives stronger information than the filling inequality for X itself. In dimension 3, we present a sufficient non-vanishing condition in terms of Massey products. This condition holds for certain manifolds that do not fiber over their Jacobi torus, such as 0-framed surgeries on suitable links. Our systolic inequality applies to surface bundles over the circle (provided the algebraic monodromy has 1-dimensional coinvariants), even though the Massey product invariant vanishes for some of these bundles.