Gr\"obner bases for (all) Grassmann manifolds
Abstract
Grassmann manifolds Gk,n are among the central objects in geometry and topology. The Borel picture of the mod 2 cohomology of Gk,n is given as a polynomial algebra modulo a certain ideal Ik,n. The purpose of this paper is to understand this cohomology via Gr\"obner bases. Reduced Gr\"obner bases for the ideals Ik,n are determined. An application of these bases is given by proving an immersion theorem for Grassmann manifolds G5,n, which establishes new immersions for an infinite family of these manifolds.
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