Babenko, I. K.; Bogatyj, S. A. Mapping a sphere into Euclidean space. (English. Russian original) Zbl 0694.55005 Math. Notes 46, No. 3, 683-686 (1989); translation from Mat. Zametki 46, No. 3, 3-8 (1989). The following hypothesis of Knaster is well known: For any mapping \(f: S^ n\to {\mathbb R}^ m\) and any \(n-m+2\) points \(\{x_ i\}_{i=1}^{n- m+2}\) of \(S^ n\) there is \(g\in \mathrm{SO}(n+1)\) such that \(f(g(x_ i))=f(g(x_ j))\), \(1\leq i,j\leq n-m+2\). Recently Makeev proved that for \(m>2\) the Knaster hypothesis is not true. In the present paper it is shown that a counterexample of Knaster’s hypothesis can be chosen to be a map defined by homogeneous polynomials. Precisely, the following theorem is proved: Let \(l(n-1)<m(C^ t_{t+l-1}+C^{t-1}_{t+l-2}-1)\). Then the set \(\{f\in H: f\) is not constant on any sphere \(S^{l-1}\subset \mathbb R^ n\}\) is open and dense in \(H\), where \(H\) is the space of all mappings from \({\mathbb R}^ n\) to \({\mathbb R}^ m\) given by polynomials with degree no more than \(t\). Reviewer: V. Valov Cited in 3 Documents MSC: 55M99 Classical topics in algebraic topology Keywords:Knaster hypothesis; homogeneous polynomials PDFBibTeX XMLCite \textit{I. K. Babenko} and \textit{S. A. Bogatyj}, Math. Notes 46, No. 3, 683--686 (1989; Zbl 0694.55005); translation from Mat. Zametki 46, No. 3, 3--8 (1989) Full Text: DOI References: [1] B. Knaster, ?Problem 4,? Colloq. Math., No. 1, 30 (1947). [2] G. R. Livesay, ?On a theorem of F. J. Dyson,? Ann. Math.,59, No. 2, 227-229 (1954). · Zbl 0056.41901 [3] H. Yamabe and Z. Yujobi, ?On the continuous functions defined on a sphere,? Osaka Math. J.,2, No. 1, 19-22 (1950). · Zbl 0039.39102 [4] S. A. Bogatyi and G. N. Khimshiashvili, ?Knaster’s problem of mapping a sphere into a line,? in: Tirapol’ Symposium on General Topology and Its Applications [in Russian], Shtiintsa, Kishinev (1985), pp. 28-30. [5] C. T. Yang, ?Continuous functions from spheres to euclidean spaces,? Ann. Math.,62, No. 2, 284-292 (1955). · Zbl 0067.15203 [6] C. T. Yang, ?On maps from spheres to euclidean spaces,? Am. J. Math.,79, No. 4, 725-732 (1957). · Zbl 0093.36001 [7] V. V. Makeev, ?Spatial generalizations of some theorems on convex figures,? Mat. Zametki,36, No. 3, 405-415 (1984). [8] A. Yu. Volovikov, ?Maps of free Z p -spaces into a manifold,? Izv. Akad. Nauk SSSR, Ser. Mat.,46, No. 1, 36-55 (1982). [9] S. S. Cairns, ?Circumscribed cubes in euclidean n-spaces,? Bull. Am. Math. Soc.,65, No. 5, 327-328 (1959). · Zbl 0087.37402 [10] S. S. Cairns, ?Umbeschreibende Wurfel von convexen Körpern in euklidischen Räumen,? J. Reine Angew. Math.,208, No. 1, 91-101 (1961). · Zbl 0106.36502 [11] V. V. Makeev, ?Properties of continuous maps of spheres and problems of combinatorial geometry,? in: Geometric Questions of the Theories of Functions and Sets [in Russian], Kalinin (1986), pp. 75-85. · Zbl 0968.52500 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.