Free circle actions on certain simply connected $7-$manifolds

In this paper, we determine for which nonnegative integers $k$, $l$ and for which homotopy $7-$sphere $\mathrm{\Sigma }$ the manifold $k{S}^2\times S^5\mathrm{\#}l{S}^3\times S^4\mathrm{\#}\mathrm{\Sigma }$ admits a free smooth circle action.