$\mathcal{C}$ will denote the collection of all connected compact (not necessarily orientable) smooth 2-dimensional surfaces-without-boundary embedded in $M$ ( here $M$ is a complete Riemannian 3-manifold). $\mathcal{C}_1$ will denote the collection of compact embedded surfaces $\Sigma$ such that each connected component of $\Sigma$ is an element of $\mathcal{C}.$
Let $\Sigma_1,\Sigma_2\in \mathcal{C}_1$ be such that $\Sigma_2$ is a $\gamma$-reduction of $\Sigma_1$.Is it true that $\Sigma_2$ has a connected component homeomorphic to $\Sigma_1$?