6. Quasipositive Curvature and Positive Curvature on an Open Dense Set

Problem 6.1.
Which theorems from $\sec > 0$ carry over to positive curvature on an open dense set? 
Problem 6.2.
Suppose $G$ is a compact Lie group with a leftinvariant metric. Are there any new examples $H \backslash G$ with quasipositive curvature? 
Problem 6.3.
Find new examples of fundamental groups in quasipositive curvature or positive curvature on an open dense set. 
Problem 6.4.
Fix $k \in \N$. Is there a $n_0 = n_0 (k)$ such that for any quasipositively curved manifold $(M^n, g)$ with $n \geq n_0$ and $\operatorname{cohom}(M^n,g) \leq k$, there exists a chain $$M_0 = M^n \subset M_1^{n+k} \subset M_2^{n+2k} \subset \cdots$$ such that all inclusions are totally geodesic, the manifolds $M_i$ are quasipositively curved, $\cup M_i$ is the classifying space of a Lie group, and $M_i / {Iso}(M_i,g)$ is isometric to $M / {Iso}(M,g)$?
Cite this as: AimPL: Manifolds with nonnegative sectional curvature, available at http://aimpl.org/nnsectcurvature.