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**Euler characteristics of collapsing Alexandrov spaces(****arXiv****)**

**Author : **Tadashi Fujioka

**Abstract :** We prove that the Euler characteristic of any collapsing Alexandrov space is equal to the sum of the products of the Euler characteristics with compact support of the strata of the limit space and the Euler characteristics of the regular fibers. This was conjectured by Semyon Alesker. We also give a proof of the existence of a constructible function on the limit space, which was asserted by Alesker as an unpublished result of Anton Petrunin.

**2.Curvature tensor of smoothable Alexandrov spaces (****arXiv****)**

**Author : **Nina Lebedeva, Anton Petrunin

**Abstract : **We prove weak convergence of curvature tensors of Riemannian manifolds for converging noncollapsing sequences with a lower bound on sectional curvature.

**3. Collapsing to Alexandrov spaces with isolated mild singularities(****arXiv****)**

**Author : **Tadashi Fujioka

**Abstract :** Let Mj be a sequence of Riemannian manifolds with sectional curvature bound below collapsing to a compact Alexandrov space X of dimension k. Suppose that all but finitely many points of X are (k,δ)-strained and that the space of directions at each exceptional point contains k+1 directions making obtuse angles with each other. We prove that Mj admits a structure of locally trivial fibration over X for sufficiently large j. The same is true for collapsing sequences of Alexandrov spaces such that the infimum of the volume of the spaces of directions is sufficiently large relative to δ.

**4.Good Coverings of Proximal Alexandrov Spaces. Homotopic Cycles in Jordan Curve Theorem Extension (****arXiv****)**

**Author : **J. F. Peters, T. Vergili

**Abstract : **This paper introduces proximal homotopic cycles, which lead to the main results in this paper, namely, extensions of the Mitsuishi-Yamaguchi Good Coverning Theorem with different forms of Tanaka good cover of an Alexandrov space equipped with a proximity relation and extensions of the Jordan curve theorem. An application of these results is also given.

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