Curves on algebraic models of smooth manifolds
type: Item
instance of: scholarly article
title: Curves on algebraic models of smooth manifolds
zbMATH Open document ID: 1170.14043
author: Wojciech Kucharz
published in: Mathematical Research Letters
publication date: 2008-08-12T00:00:00Z
review text: Every smooth compact manifold \(M\) is diffeomorphic to some nonsingular real algebraic set \(X\), which is called an algebraic model of \(M\). Algebraic models are not unique, cf. \textit{J. Bochnak, M. Coste} and \textit{M.-F. Roy} [Real Algebraic Geometry, Springer, Berlin (1998; Zbl 0912.14023)], (14.1.13). The paper relates subgroups of the homology group \(H_1(M,\mathbb{Z}/2)\) to subgroups of the homology groups \(H_1(X, \mathbb{Z}/2)\) as \(X\) varies in the set of algebraic models of \(M\). The homology classes of irreducible algebraic curves on \(X\) generate a subgroup \(H_1^{\text{alg}}(X,\mathbb{Z}/2)\leq H_1(X,\mathbb{Z}/2)\). Inside \(H_1^{\text{alg}} (X,\mathbb{Z}/2)\) there are the subgroups \(A(X,g)\) and \(B(X,g)\) generated by the homology classes of curves of genus exactly \(g\) and of genus at most \(g\). An algebraic curve \(C\) on \(X\) is dividing if the complexification without the real points is disconnected. The dividing real algebraic curves generate a subgroup \(D(X)\leq H_1^{\text{alg}}(X, \mathbb{Z}/2)\). It is always true that \(A(X,0)\leq D(X)\). The author proves the following Theorem. If \(M\) is a smooth compact manifold of dimension at least 2, if \(g\in\mathbb{N}\) and if \(G\leq H_1(M,\mathbb{Z}/2)\) is a subgroup, then there exists an algebraic model \(\varphi: M\to X\) such that \(A(X,g)=B(X,g)\) and the isomorphism \(\varphi_1:H_1(M,\mathbb{Z}/2) \to H_1(X,\mathbb{Z}/2)\) maps \(G\) onto \(A(X,g)\). With additional hypotheses, a similar result is proved for a prescribed subgroup \(G\leq H_1(M,\mathbb{Z}/2)\) and the subgroups \(A(X,0) \leq D(X)\leq H_1(X,\mathbb{Z}/2)\).
reviewed by: Niels Schwartz
Mathematics Subject Classification ID: 14P25
Mathematics Subject Classification ID: 57R19
Mathematics Subject Classification ID: 14H45
Mathematics Subject Classification ID: 14P05
zbMATH DE Number: 5310632
zbMATH Keywords: smooth manifold
zbMATH Keywords: algebraic variety
zbMATH Keywords: algebraic curve
zbMATH Keywords: homology group
zbMATH Keywords: algebraic homology groups
MaRDI profile type: Publication
full work available at URL: mrl.2008.v15.n2.a7
OpenAlex ID: W1980639792
DOI: 10.4310/MRL.2008.V15.N2.A7
description: scientific article; zbMATH DE number 5310632