  We give inequalities for the Manolescu invariants $\alpha,\beta,\gamma$ under
the connected sum operation. We compute the Manolescu invariants of connected
sums of some Seifert fiber spaces. Using these same invariants, we provide a
proof of Furuta's Theorem, the existence of a $\mathbb{Z}^\infty$ subgroup of
the homology cobordism group. To our knowledge, this is the first proof of
Furuta's Theorem using monopoles. We also provide information about Manolescu
invariants of the connected sum of $n$ copies of a three-manifold $Y$, for
large $n$.
