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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.