Min-max minimal hypersurface in $(M^{n+1},g)$ with $Ri{c}_g>0$ and $2\le n\le 6$

In this paper, we study the shape of the min-max minimal hypersurface produced by Almgren-Pitts in \cite{A2}\cite{P} corresponding to the fundamental class of a Riemannian manifold $(M^{n+1},g)$ of positive Ricci curvature with $2\le n\le 6$. We characterize the Morse index, area and multiplicity of this min-max hypersurface. In particular, we show that the min-max hypersurface is either orientable and of index one, or is a double cover of a non-orientable minimal hypersurface with least area among all closed embedded minimal hypersurfaces.