The Borsuk-Ulam theorem for planar polygon spaces. (English) Zbl 1496.55002
The moduli space of planar polygons with prescribed generic side lengths is a smooth manifold equipped with a natural fixed point free involution. So a Borsuk-Ulam-type question makes sense. As it is known, this question is related with \(\mathbb{Z }_2\)-index, coindex and Stiefel-Whitney height. The paper contains the computations and estimations of these numbers.
The topology of the moduli space depends on the side lengths, namely, on the genetic code (some combinatorial data). A formula for the Stiefel-Whitney height of the moduli space in terms of the genetic code is obtained.
The topology of the moduli space depends on the side lengths, namely, on the genetic code (some combinatorial data). A formula for the Stiefel-Whitney height of the moduli space in terms of the genetic code is obtained.
Reviewer: Gaiane Panina (St. Petersburg)
MSC:
| 55M20 | Fixed points and coincidences in algebraic topology |
| 55M30 | Lyusternik-Shnirel’man category of a space, topological complexity à la Farber, topological robotics (topological aspects) |
| 55P15 | Classification of homotopy type |
| 57R42 | Immersions in differential topology |
| 55R80 | Discriminantal varieties and configuration spaces in algebraic topology |
Keywords:
free \(\mathbb{Z}_2\)-space; \(\mathbb{Z }_2\)-coindex; \(\mathbb{Z }_2\)-index; Stiefel-Whitney class; tidy spaces; planar polygon spacesReferences:
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