On an invariant curvature cone along 4-dimensional Ricci flow

In this paper, we study 4-dimensional complete non-compact manifold with its curvature operator in $\mathfrak{C}_{\eta,\mu}$ via Ricci flow. We obtain topological and geometric gap theorems assuming such manifold has maximal volume growth. We also study 4-dimensional complete manifold with lower bound of $\mathfrak{C}_{\eta,\mu}$ and obtain regularity results for Gromov-Hausdorff limit of complete volume non-collapsed manifolds with lower bound of $\mathfrak{C}_{\eta,\mu}$.