Examples of algebraically realized maps
type: Item
instance of: scholarly article
title: Examples of algebraically realized maps
zbMATH Open document ID: 1366.14052
author: Karl Heinz Dovermann
author: Jason Hanson
author: Robert D. Little
published in: Geometriae Dedicata
publication date: 2017-02-17T00:00:00Z
review text: This paper discusses the following conjecture: Let \(G\) be a compact Lie group. Then every closed smooth \(G\)-manifold is equivariantly diffeomorphic to a nonsingular real algebraic \(G\)-variety \(X\) so that every \(G\) vector bundle over \(X\) is strongly algebraic, i.e., classified by an entire rational map. This algebraic realization problem can be reduced to a bordism problem. The paper under review deals with this bordism problem for the case \(G=\mathbb{Z}_4\). The authors construct examples of \(G\)-equivariant entire rational maps from non-singular real algebraic \(G\)-varieties to Grassmannians with appropriate actions. It is shown that these maps represent all classes in certain equivariant bordism groups of the Grassmannians. Using these examples one can prove the conjecture for \(G=\mathbb{Z}_4\). Moreover, in the preprint [\textit{K. H. Dovermann} and \textit{A. G. Wasserman}, ``Algebraic realization of cyclic group actions'', \url{http://www.math.hawaii.edu/~heiner/cbonn.pdf}] these examples are used to prove the above conjecture for all cyclic groups \(G\).
reviewed by: Michael Wiemeler
Mathematics Subject Classification ID: 14P25
Mathematics Subject Classification ID: 57S15
Mathematics Subject Classification ID: 57S25
Mathematics Subject Classification ID: 14M15
Mathematics Subject Classification ID: 55N22
zbMATH DE Number: 6686307
zbMATH Keywords: algebraic models
zbMATH Keywords: equivariant bordism
MaRDI profile type: Publication
full work available at URL: s10711-016-0177-x
OpenAlex ID: W2434596826
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DOI: 10.1007/S10711-016-0177-X
description: scientific article; zbMATH DE number 6686307