The Borsuk-Ulam property for maps from the product of two surfaces into a surface
Let \(X\) and \(Y\) be topological spaces and \(\alpha\) a fixed-point free involution on \(X\). The triple \((X, \alpha, Y)\) is said to satisfy the Borsuk-Ulam property if for each continuous map \(f: X \to Y\) there exists a point \(x \in X\) such that \(f(\alpha(x))=f(x)\). The paper under review considers a special case of this situation. Let \(\alpha_1\) and \(\alpha_2\) be free involutions on closed connected surfaces \(S_1\) and \(S_2\), respectively. Let \(\alpha_1 \times \alpha_2\) be the induced diagonal involution on the 4-manifold \(S_1 \times S_2\). Given another closed connected surface \(S\), the authors determine a necessary and sufficient condition under which the triple \((S_1 \times S_2, \alpha_1 \times \alpha_2, S)\) satisfies the Borsuk-Ulam property. More precisely, the main result in this nicely written paper is the following theorem. \par Theorem 1.1. The triple \((S_1 \times S_2, \alpha_1 \times \alpha_2; S)\) satisfies the Borsuk-Ulam property if and only if the triples \((S_1,\alpha_1;S)\) and \((S_2,\alpha_2;S)\) satisfy the Borsuk-Ulam property. The problem can also be formulated in terms of an algebraic diagram involving the 2-string braid group \(B_2(S)\).
scientific article; zbMATH DE number 7474119
scientific article