Moduli Spaces of Positive Curvature Metrics in Dimension Four and Beyond

We construct non-trivial elements in the homotopy groups of the observer moduli space of positive sectional curvature metrics on $\mathbb{C}P^n$ and non-trivial elements in the homotopy groups of the observer moduli space of positive scalar curvature metrics on $\mathbb{C}P^2 \sharp M^4$.