Formal and non-formal homogeneous spaces of small rank.

Abstract

The aim of this thesis is to determine which 2-tori T makes $G/T$ formal for a compact connected Lie group G of rank 3. We show that the only time there is a possibility of a non-formal homogeneous space $G/T$ is when the Lie algebra E of G is semisimple and contains three simple ideals. In such a case, the Koszul complex is given by$$ (\Lambda(y\sb2,z\sb2,x\sb3,x\sbsp{3}{\prime}, x\sbsp{3}{\prime\prime}),d)$$and$$dx\sb3 = -y\sbsp{2}{2},\ dx\sbsp{3}{\prime} = -z\sbsp{2}{2},\ {\rm and}\ dx\sbsp{3}{\prime\prime} = -(\alpha y\sb2 + \beta z\sb2)\sp2,$$where $\alpha,\beta\in$ Q. We proveTheorem 5.5.4: This minimal c.g.d.a is formal if and only if $\alpha=0$ or $\beta=0.$ This, as we will see, indicates, in the case of non-formality, a special mixing of the 2-torus inside G.