Mapping theorems for cohomologically trivial maps
Some mapping theorems for maps of the n-sphere S^n are obtained. As a corollary, it is shown that every cohomologically trivial map f:S^n on→ Y of S^n onto some Y identifies a pair of points x_1,x_2 S^n such that the distance between them is not less than the diameter of the regular (n+1)-simplex inscribed in S^n: f(x_1)=f(x_2), 0.2cm ||x_1-x_2||≥ 2(n+2)n+1 Furthermore, it is proved that for any decomposition of S^n into n closed subsets some of them contains a continuum K with diam k ≥ 2(n+2)n+1 . Also it is shown that every lowering dimension map f:S^n → Y is constant on a continuum K with diam K ≥ 2(n+2)n+1 . Finally, a mapping theorem for maps of S^n into k-dimensional contractible polyhedra is obtained (for k < n) , journal=Ann. Sofia Univ. Fac. Math. and Inf., author=Stefanov, Simeon, year=1995, month=Dec., pages=287–295