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 $f:S^n \overset{on}{\rightarrow} Y$ 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 $S^n$:

$f(x_1)=f(x_2), \hspace{0.2cm} ||x_1-x_2||\geq \sqrt{\frac{2(n+2)}{n+1}}$

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