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

TL;DR

This paper investigates the properties of Johnson's Abel-Jacobi invariants for the Torelli group, revealing their non-injectivity, surjectivity in certain cases, and implications for the group's homology, with conjectures on stability.

Contribution

Introduces methods to compute Johnson's invariants tau_i, proves non-injectivity for i > 1, and demonstrates the infinite-dimensionality of Torelli group homology.

Findings

tau_i is not injective for 2 <= i < g

tau_2 is surjective

H_i(I_g,*) is nonzero for 1 <= i < g

Abstract

Let I_g,* denote the (pointed) Torelli group. This is the group of homotopy classes of homeomorphisms of the genus g >= 2 surface S_g with a marked point, acting trivially on H := H_1(S_g). In 1983 Johnson constructed a beautiful family of invariants tau_i: H_i(I_g,*) -> /\^{i+2} H for 0 <= i <= 2g-2, using a kind of Abel-Jacobi map for families, in order to detect nontrivial cycles in I_g,*. Johnson proved that tau_1 is an isomorphism rationally, and asked if the same is true for tau_i with i > 1. The goal of this paper is to introduce various methods for computing tau_i; in particular we prove that tau_i is not injective (even rationally) for any 2 <= i < g, and that tau_2 is surjective. For g >= 3, we find enough classes in the image of tau_i to deduce that H_i(I_g,*, Q) is nonzero for each 1 <= i < g, in contrast with mapping class groups. Many of our classes are stable, so we can deduce that H_i(I_infty,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.