arXiv subjects

Nikolai Mnëv

Publications and source records attributed to Nikolai Mnëv.

3 recordsLinked to original sources

arXiv2024

$K(Z,2)$ out of circular permutations

We discuss $\pmb{SC}_*$, a simplicial homotopy model of $K(Z,2)$ constructed from circular permutations. In any dimension, the number of simplices in the model is finite. The complex $\pmb{SC}_*$ naturally manifests as a simplicial set representing ``minimally" triangulated circle bundles over simplicial bases. On the other hand, existence of the homotopy equivalence $|\pmb{SC}_*| \approx B(U(1)) \approx K(Z,2)$ appears to be a canonical fact from the foundations of the theory of crossed simplicial groups.

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arXiv2021

A note on a local combinatorial formula for the Euler class of a PL spherical fiber bundle

We present a local combinatorial formula for the Euler class of a $n$-dimen\-si\-onal PL spherical fiber bundle as a rational number $e_{\it CH}$ associated to a chain of $n+1$ abstract subdivisions of abstract $n$-spherical PL cell complexes. The number $e_{\it CH}$ is a combinatorial (or matrix) Hodge theory twisting cochain in Guy Hirsch's homology model of the bundle associated with PL combinatorics of the bundle.

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arXiv2019

Minimal triangulations of circle bundles, circular permutations and binary Chern cocycle

We investigate a PL topology question: which circle bundles can be triangulated over a given triangulation of the base? The question got a simple answer emphasizing the role of minimal triangulations encoded by local systems of circular permutations of vertices of the base simplices. The answer is based on an experimental fact: classical Huntington transitivity axiom for cyclic orders can be expressed as the universal binary Chern cocycle.

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