ABELIAN CYCLES IN THE HOMOLOGY OF THE TORELLI GROUP

In the early 1980's, Johnson defined a homomorphism $\mathcal{I}_{g}^1\to\bigwedge^3 H_1(S_{g},\mathbb{Z})$, where $\mathcal{I}_{g}^1$ is Torelli group of closed, connected and oriented surface genus $g$ with boundary component $S_g$ corresponding without component. This known as homomorphism. We study map induced by on rational homology groups apply it to abelian cycles determined disjoint bounding pair maps, in order compute large quotient $H_n(\mathcal{I}_{g}^1,\mathbb{Q})$ stable range. also implies an analogous result for $\mathcal{I}_{g,1}$ marked point instead Further, we investigate how much image this generated images such use prove that pointed case, they generate proper subrepresentation $H_n(\mathcal{I}_{g,1})$ $n\ge 2$ enough.