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On local combinatorial formulas for Euler class of spherical fiber bundle.

  • Speaker:Dr. Nikolai Mnev, Senior Research Fellow of Chebyshev Laboratory at Saint-Petersburg State University and Saint Petersburg Department of Steklov Mathematical Institute
  • TIME:周四20:00-21:00,2021-01-28
  • LOCATION:online

Beijing-Saint Petersburg Mathematics Colloquium (online)

Abstract 

We will discuss the classical problem on local combinatorial formulas of characteristic classes on example of Euler class. Suppose we have a PL spherical fiber bundle with a fiber S^n   triangulated over the base simplicial complex.  The  bundle determines  n+1 dimensional Euler characteristic class in the base. Local combinatorial formula for the Euler class is a universal combinatorial  function of elementary triangulated  S^n-bundles over n+1 simplices universally representing Euler cocycle of the bundle in simplicial cohomology of the base.   Such  functions exist for rational coefficients in cohomology.   They can be constructed  as   "twisting cochains" --  explicit local chain-level formulas for Gysin homomorphism  in the Gysin sequence of the bundle.  To get an access  to local chain combinatorics of spectral sequence of the bundle we may use Guy Hirsh  homology model of the bundle as a local system and then applying homology perturbation theory obtain local formulas as certain measure of twisting in combinatorial Hodge  structure of the elementary bundle.   The answer can be interpreted and evaluated statistically as certain combinatorial counting using  Catanzaro-Chernyak-Klein higher  Kirchhoff theorems.   The formulas are resulting in a combinatorial form of Gauss-Bonne theorem. For example  we easily obtain otherwise difficult to access statement: One can triangulate only trivial and Hopf circle bundles over a 2-dimensional sphere if the base sphere is triangulated as the boundary of 3-simplex.

Bio

Dr. Nikiolay Mnev graduated from the faculty of Mathematics and Mechanics of Saint Petersburg State University in 1980. In 1986 he defended his PhD thesis under supervision of Prof. Anatoly Vershik. Since that time he works at Steklov Institute and, besides, he has got a position at Chebyshev Laboratory starting from its foundation in 2011. Additionally, Dr. Mnev curates the so-called Fizmatklub, the organization intended to boost the level of teaching maths in Saint Petersburg Universities by organizing additional lecture courses, delivered by leading specialists.

Dr. Nikolai Mnev’s research interests cover Algebraic Topology, Geometric Topology and Combinatorial Geometry. In 1991, he received Delbert Ray Fulkerson prize for outstanding papers in the area of discrete mathematics.

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