Mapping theorems for cohomologically trivial maps

Abstract

Some mapping theorems for maps of the $n$-sphere $S^n$ are obtained. As a corollary, it is shown that every cohomologically trivial map $f:S^n\stackrel{on}{\to }Y$ of $S^n$ onto some $Y$ identifies a pair of points $x_1,x_2\in S^n$ such that the distance between them is not less than the diameter of the regular $(n+1)$-simplex inscribed in $S^n$:

$f(x_1)=f(x_2),||x_1-x_2||\ge \sqrt{\frac{2(n+2)}{n+1}}$

Furthermore, it is proved that for any decomposition of $S^n$ into $n$ closed subsets some of them contains a continuum $K$ with diam $k\ge \sqrt{\frac{2(n+2)}{n+1}}$. Also it is shown that every lowering dimension map $f:S^n\to Y$ is constant on a continuum $K$ with diam $K\ge \sqrt{\frac{2(n+2)}{n+1}}$. Finally, a mapping theorem for maps of $S^n$ into $k$-dimensional contractible polyhedra is obtained (for $k<n$)