Lf-index bounds for closed f-minimal hypersurfaces

We discuss a two-side closed embedded f-minimal hypersurface Σ in a smooth metric measure space (Mn+1,g¯,e−fdμ) and obtain that Lf-index of Σ is bounded from below by a constant times the first Betti number with a suitable condition. We also apply the result to f-minimal hypersurfaces in shrinking gradient solitons.