2. Collapse and Alexandrov Geometry

Problem 2.05.
Perelman’s Stability Theorem yields that manifolds in a given sequence of noncollapsing manifolds are eventually pairwise homeomorphic. Are they also PLhomeomorphic or diffeomorphic? 
Problem 2.1.
Understand DCstructures on manifolds. In particular, does Perelman’s Stability Theorem hold in the DCcategory. Is PL = DC always? 
Problem 2.15.
Extend the Wilking Connectivity Theorem to Alexandrov spaces, i.e. if $X$ is a positively curved Alexandrov space and $Y \subset X$ a totally geodesic subspace of codimension $k$, is it true that $XY$ has homology only up to dimension $2k  2$? 
Problem 2.2.
Suppose $X$ is the noncollapsed GromovHausdorff limit of $(M_i^n, g_i)$, where $\sec_{M_i} \leq 1$, and that along every geodesic on $M_i$ one hits a conjugate point before $t = \pi + \frac{1}{i}$. Is $X$ rigid in any sense? 
Problem 2.25.
Is there a sequence of simplyconnected, pointwise strictly $\frac{1}{4}$pinched manifolds $M_i^n$, $n > 2$, that collapse? 
Problem 2.3.
Find an appropriate definition of Morse functions on Alexandrov spaces and construct examples. 
Consider finite towers \begin{equation} \xymatrix{ M_0 \ar[r]^{F_1} & M_1 \ar[r]^{F_2} & \dots \ar[r]^{F_k} & M_k } \end{equation} of fiber bundles, where the fibers $\{F_1, \dots, F_k\}$ and $M_k$ are fixed topological manifolds.
Problem 2.4.
Loosen this notion to get a “brotherhood” on the manifolds $M_0$ and a property of such $M_0$ not known to be possessed by all manifolds of $\sec \geq K$, $\operatorname{diam} \leq 1$. 
Problem 2.45.
Give an Alexandrov analogue of rational ellipticity. In particular, are manifolds with almost nonnegative sectional curvature rationally elliptic? 
Problem 2.5.
Given a noncollapsing GromovHausdorff convergence $M_i \longrightarrow X$, can one find a “tangent bundle” structure on $X$ that is sensitive to the diffeomorphism class of the $M_i$? 
Problem 2.55.
Is there a Gauss formula for Alexandrov spaces, i.e. must a convex hypersurface $Y$ of an Alexandrov space $X$ have $\sec_Y \geq \sec_X$? 
Problem 2.6.
Is every finite dimensional Alexandrov space a limit of Riemannian manifolds with $\sec \geq K$? 
Problem 2.7.
Is there an alternate approach to homotopy groups that is adapted to Alexandrov spaces? 
Problem 2.8.
Can an $n$dimensional torus collapse to an interval? 
Problem 2.85.
Study the collapse of Riemannian manifolds with boundary which have $\sec \geq K$ on the interior and controlled boundary concavity. 
Problem 2.9.
Find an application where infinitedimensional Alexandrov spaces appear as limits of manifolds of increasing dimension.
Cite this as: AimPL: Manifolds with nonnegative sectional curvature, available at http://aimpl.org/nnsectcurvature.