Parameterized Abel-Jacobi maps and abelian cycles in the Torelli group

Abstract

Let ℐg, * denote the Torelli group of the genus g⩾2 surface Sg with one marked point. This is the group of homotopy classes (rel basepoint) of homeomorphisms of Sg fixing the basepoint and acting trivially on H≔ H1(Sg, ℚ). In 1983, Johnson constructed a beautiful family of invariants τ i : H i ( ℐ g , * , ℚ ) → ∧ i + 2 H , for 0⩽i⩽2g−2, using a kind of Abel–Jacobi map for families. He used these invariants to detect non-trivial cycles in ℐg, *. Johnson proved that τ1 is an isomorphism, and asked if the same is true for τi with i>1. The goal of this paper is to introduce various methods for computing τi; in particular, we prove that τi is not injective for any 2⩽i<g, answering Johnson's question in the negative. We also show that τ2 is surjective. For g⩾3 we find many classes in the image of τi and use them to deduce Hi(ℐg, *, ℚ)≠0 for each 1⩽i<g. This is in contrast to the case of mapping class groups. Many of our classes are stable, so we can deduce that Hi(ℐ∞, 1, ℚ) is infinite-dimensional for each i⩾1. Finally, we conjecture a new kind of ‘representation-theoretic stability’ for the homology of the Torelli group, for which our results provide evidence.