Min-max theory and minimal surfaces with prescribed genus

We establish a general min-max type theorem that produces minimal surfaces with prescribed genus in 3-manifolds with positive Ricci curvature. An important intermediate step is to show that, in a generic metric with positive Ricci curvature, any family of smooth embedded surfaces, possibly with finitely many singularities, can be deformed into a certain topologically optimal family. Results in this paper will be crucial to our program on the construction of multiple minimal surfaces with prescribed genus in 3-spheres via topological methods.