On homotopy groups of real algebraic varieties and their complexifications (Q1768256)

Let \(X_0\) be a connected component of a compact nonsingular real algebraic variety \(X\) and let \(i: X\to X_{\mathbb{C}}\) be a complexification of \(X\). The author studies the homomorphism \(i_{\#}: \pi_k(X_0)\to \pi_k(X_{\mathbb{C}})\) induced by \(i\). For \(k= 1,2\) he proves its independence of the chosen complexification, and so its kernel is an isomorphism invariant of \(X\). This is not longer true for \(k> 2\). Let \[ b_i= \dim_{\mathbb{Q}}(H^i(X_0, \mathbb{Q})); \quad \widetilde b_i= \dim_{\mathbb{Q}}(\text{Im\,}H^i(X_0, \mathbb{Q})), \] and let \(\varepsilon\) be the dimension of the image of the cup product evaluation map \[ \cup: P_2[\text{Im\,}H^2(X_0, \mathbb{Q})]\to H^4(X_0, \mathbb{Q}), \] where \(P_2[\text{Im\,}H^2(X_0, \mathbb{Q})]\) is the \(\mathbb{Q}\)-vector space of quadratic polynomials in \(\text{Im\,}H^2(X_0, \mathbb{Q})\). The main result of the article under review is Theorem 3.1 which establishes the inequalities \[ \widetilde b_3\leq \dim_{\mathbb{Q}}(i_{\#}(\pi_3(X_0\otimes\mathbb{Q})))\leq b_3+ {\widetilde b_2\over 2} (1+\widetilde b_2)- \varepsilon \] in case both \(X_0\) and \(X_{\mathbb{C}}\) are simply connected. The paper contains also many interesting examples.