Curves on algebraic models of smooth manifolds
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 \(A(X,0) \leq D(X)\leq H_1(X,\mathbb{Z}/2)\).
scientific article; zbMATH DE number 5310632