Topology of real algebraic threefolds
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)].
affine nonsingular real algebraic threefold
homology classes which can be represented by algebraic subsets
scientific article; zbMATH DE number 4024321
scientific article