Gonçalves, Daciberg Lima; dos Santos, Anderson Paião; Silva, Weslem Liberato The Borsuk-Ulam property for maps from the product of two surfaces into a surface. (English) Zbl 1491.55002 Topol. Methods Nonlinear Anal. 58, No. 2, 367-388 (2021). 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. 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 \(B_2(S)\). Reviewer: Mahender Singh (Sahibzada Ajit Singh Nagar) Cited in 2 Documents MSC: 55M20 Fixed points and coincidences in algebraic topology 55M35 Finite groups of transformations in algebraic topology (including Smith theory) Keywords:Borsuk-Ulam theorem; involutions; surface braid groups; surface × Cite Format Result Cite Review PDF Full Text: DOI References: [1] K. Borsuk, Drei Satze uber die \(n\)-dimensionale euklidische Sphare, Fund. Math. 20 (1933), 177-190. · JFM 59.0560.01 [2] E. Fadell and S. Husseini, The Nielsen number on surfaces, Contemp. Math. 21 (1983), 59-98. · Zbl 0563.55001 [3] D.L. Goncalves and J. Guaschi, The Borsuk-Ulam theorem for maps into a surface, Topology Appl. 157 (2010), 1742-1759. · Zbl 1194.55005 [4] D.L. Goncalves, J. Guaschi and V.C. Laass, The Borsuk-Ulam property for homotopy classes of selfmaps of surfaces of Euler characteristic zero, J. Fixed Point Theory Appl. (2019), 21-65. · Zbl 1418.55001 [5] D.L. Goncalves, C. Hayat and P. Zvengrowski, The Borsuk-Ulam theorem for manifolds, with applications to dimensions two and three, Groups Actions and Homogeneous Spaces, Fak. Mat. Fyziky Inform. Univ. Komenskeko, Bratislava, 9-28, 2010. · Zbl 1220.55001 [6] D.L. Goncalves and A.P. Santos, Diagonal involutions and the Borsuk-Ulam property for product of surfaces, Bull. Braz. Math. Soc. (N.S.) 50 (2019), no. 3, 771-786. · Zbl 1426.55002 [7] D.L. Johnson, Presentation of Groups, London Math. Soc. Lecture Note Ser., vol. 22, Cambrige Uni. Press, 1976. · Zbl 0324.20040 [8] W. Magnus, A. Karrass and D. Solitar, Combinatorial Group Theory: presentations of groups in terms of generators and relations, Pure and Applied Mathematics: A series of texts and monographs, vol. XIII, Interscience Publishers, 1966. · Zbl 0138.25604 [9] T. tom Dieck, Transformation Groups, Walter de Gruyter, Berlin, New York, 1987 · Zbl 0611.57002 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.