Семинар по геометрической топологии. Докладчик : Андрей Рябичев

Suppose we are given smooth manifolds M,N and a continuous map f:M\to N. We may ask, when is f homotopic to a smooth map with a prescribed singular locus? The case of fold singularities was settled by Y.Eliashberg in the 1970s. Namely, there is a necessary and sufficient condition for f to be homotopic to a smooth map with prescribed folds C\subset M and with no other critical points. We will discuss how one can generalize this condition for an arbitrary locus of Thom-Boardman singularities.

The most well-known case is the manifold of isospectral tridiagonal matrices. This manifold is closely related to the toric variety of type A known in representation theory. This relation can be extended to the relation between manifolds of isospectral staircase matrices and semisimple regular Hessenberg varieties: they have homeomorphic orbit spaces and isomorphic equivariant cohomology rings. 

We study two more examples: the manifold of arrow matrices, and the space of periodic tridiagonal matrices. The study of topology in these two examples had lead us to surprisingly interesting objects from combinatorial geometry: the maximal cubical subcomplex of a permutohedron and the regular tiling of Euclidean space by permutohedra. 

The talk is partly based on joint works with Victor Buchstaber.

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