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2. Collapse and Alexandrov Geometry


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      Problem 2.05.

      Perelman’s Stability Theorem yields that manifolds in a given sequence of non-collapsing manifolds are eventually pairwise homeomorphic. Are they also PL-homeomorphic or diffeomorphic?


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          Problem 2.1.

          Understand DC-structures on manifolds. In particular, does Perelman’s Stability Theorem hold in the DC-category. Is PL = DC always?


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              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 $X-Y$ has homology only up to dimension $2k - 2$?


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                  Problem 2.2.

                  Suppose $X$ is the non-collapsed Gromov-Hausdorff 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?


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                      Problem 2.25.

                      Is there a sequence of simply-connected, pointwise strictly $\frac{1}{4}$-pinched manifolds $M_i^n$, $n > 2$, that collapse?


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                          Problem 2.3.

                          Find an appropriate definition of Morse functions on Alexandrov spaces and construct examples.


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                              Problem 2.35.

                              Study the collapse of Alexandrov spaces.

                                •     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.

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                                  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 $\sec \geq K$, $\operatorname{diam} \leq 1$.


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                                      Problem 2.45.

                                      Give an Alexandrov analogue of rational ellipticity. In particular, are manifolds with almost non-negative sectional curvature rationally elliptic?


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                                          Problem 2.5.

                                          Given a non-collapsing Gromov-Hausdorff convergence $M_i \longrightarrow X$, can one find a “tangent bundle” structure on $X$ that is sensitive to the diffeomorphism class of the $M_i$?


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                                              Problem 2.55.

                                              Is there a Gauss formula for Alexandrov spaces, i.e. must a convex hypersurface $Y$ of an Alexandrov space $X$ have $\sec_Y \geq \sec_X$?


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                                                  Problem 2.6.

                                                  Is every finite dimensional Alexandrov space a limit of Riemannian manifolds with $\sec \geq K$?


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                                                      Problem 2.65.

                                                      Study Alexandrov (almost) submetries.


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                                                          Problem 2.7.

                                                          Is there an alternate approach to homotopy groups that is adapted to Alexandrov spaces?


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                                                              Problem 2.75.

                                                              Study collapse to a ray.


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                                                                  Problem 2.8.

                                                                  Can an $n$-dimensional torus collapse to an interval?
                                                                      (The answer to this question is “No”, essentially settled at the workshop.)


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                                                                      Problem 2.85.

                                                                      Study the collapse of Riemannian manifolds with boundary which have $\sec \geq K$ on the interior and controlled boundary concavity.


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                                                                          Problem 2.9.

                                                                          Find an application where infinite-dimensional Alexandrov spaces appear as limits of manifolds of increasing dimension.

                                                                              Cite this as: AimPL: Manifolds with non-negative sectional curvature, available at http://aimpl.org/nnsectcurvature.