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In this paper, we determine the $0^\text{th}$ stable cohomotopy groups of$k$-fold connected sums of complex projective $n$-spaces $\#_k\mathbb{C}P^{n}$,within the range of $3\leq n \leq 8$. In addition, we compute the reduced $K$and $KO$-groups of $\#_k\mathbb{C}P^n$ for all $k,n\geq 2$. As a consequence,we establish the set of smooth normal invariants for these manifolds.

Stable cohomotopy and $KO$ groups of the $k$-fold connected sum of complex projective spaces