Suspension Splitting and Cohomotopy Sets of Simply Connected $7$-manifolds

Let $M$ be a closed simply connected $7$-manifold. In this paper we establish homotopy decompositions of the reduced suspension space $\mathrm{\Sigma }M$ into a wedge sum of simpler spaces when localized at a set of primes. These decompositions are applied to study the cohomotopy sets ${\pi }^k(M)$ and the $p$-local cohomotopy sets ${\pi }^4(M;{\mathbb{Z}}_{(p)})$.