Topology of real algebraic threefolds
type: Item
instance of: scholarly article
title: Topology of real algebraic threefolds
zbMATH Open document ID: 0629.57021
DOI: 10.1215/S0012-7094-86-05352-4
author: Wojciech Kucharz
published in: Duke Mathematical Journal
publication date: 1986-01-01T00:00:00Z
review text: Let X be an affine nonsingular real algebraic threefold; \(H_*(X, {\mathbb{Z}}/2 {\mathbb{Z}})\) denotes the homology of X built on infinite locally finite chains, \(H_*^{alg}(X, {\mathbb{Z}}/2 {\mathbb{Z}})\) is the subgroup of those homology classes which can be represented by algebraic subsets of X, and \(H^*_{alg}(X, {\mathbb{Z}}/2 {\mathbb{Z}})\) is the subgroup of the cohomology group corresponding to \(H_*^{alg}(X, {\mathbb{Z}}/2 {\mathbb{Z}})\) via Poincaré duality. The author shows that if X is orientable as a \(C^{\infty}\)-manifold, then (1) every homology class \(u\in H_ 1^{alg}(X, {\mathbb{Z}}/2 {\mathbb{Z}})\) can be represented by a nonsingular algebraic subset of X; if, moreover, X is compact, then (2) for each compact 1-dimensional \(C^{\infty}\)- submanifold M of X whose homology class is in \(H_ 1^{alg}(X, {\mathbb{Z}}/2 {\mathbb{Z}})\), there exists a \(C^{\infty}\)-embedding \(h: M\to X\), arbitrarily close to the inclusion \(M\hookrightarrow X\) in the \(C^{\infty}\)-topology, such that h(M) is a nonsingular algebraic subset of X; and (3) a continuous real vector bundle of constant rank over X is \(C^ 0\)-isomorphic to a strongly algebraic vector bundle if and only if its first and second Stiefel-Whitney characteristic classes belong to \(H^ 1_{alg}(X, {\mathbb{Z}}/2 {\mathbb{Z}})\) and \(H^ 2_{alg}(X, {\mathbb{Z}}/2 {\mathbb{Z}})\), respectively. The proofs rely on some of the author's earlier results [Invent. Math. 82, 19-25 (1985; Zbl 0547.14018), and Compos. Math. 60, 209-225 (1986; Zbl 0615.14007)]; for a discussion of strongly algebraic vector bundles, see the paper of \textit{R. Benedetti} and \textit{A. Tognoli} [Bull. Sci. Math., II. Sér. 104, 89-112 (1980; Zbl 0421.58001)].
reviewed by: J.Weinstein
Mathematics Subject Classification ID: 57R95
Mathematics Subject Classification ID: 55N35
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 14J30
Mathematics Subject Classification ID: 14Pxx
zbMATH DE Number: 4024321
zbMATH Keywords: Stiefel-Whitney classes
zbMATH Keywords: affine nonsingular real algebraic threefold
zbMATH Keywords: homology
zbMATH Keywords: infinite locally finite chains
zbMATH Keywords: homology classes which can be represented by algebraic subsets
zbMATH Keywords: continuous real vector bundle
zbMATH Keywords: strongly algebraic vector bundle
MaRDI profile type: Publication
cites work: The topology of real algebraic sets
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cites work: Q3854334
cites work: Q3963140
cites work: La classe d'homologie fondamentale d'un espace analytique
cites work: Projective Modules as Fiber Bundles
cites work: Vector bundles over spheres are algebraic
cites work: Algebraic vector bundles on projective space
cites work: Resolution of singularities of an algebraic variety over a field of characteristic zero. I
cites work: On homology of real algebraic sets
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cites work: Algebraic vector bundles over the 2-sphere
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description: scientific article; zbMATH DE number 4024321