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.