Title: $T\sp{3}$-actions on simply connected $6$-manifolds. I
Abstract: We are concerned with ${T^3}$-actions on simply connected 6-manifolds ${M^6}$. As in the codimension two case, there exists, under certain restrictions, a cross-section. Unlike the codimension two case, the orbit space need not be a disk and there can be finite stability groups. C. T. C. Wall has determined (Invent. Math. 1 (1966), 355-374) a complete set of invariants for simply connected 6-manifolds with ${H_\ast }({M^6})$ torsion-free and ${\omega _2}({M^6}) = 0$. We establish sufficient conditions for these two requirements to be met when M is a ${T^3}$-manifold. Using surgery and connected sums, we compute the invariants for manifolds satisfying these conditions. We then construct a ${T^3}$-manifold ${M^6}$ with invariants different than any well-known manifold. This involves comparing the trilinear forms (defined by Wall) for two different manifolds.