Universality of maps on suspensions over products of span zero continua (Q2862659)

A \textit{continuum} is a connected compact Hausdorff space; maps are always assumed to be continuous. A map \(f:X\to Y\) between continua is \textit{universal} if for every map \(g:X\to Y\), there is a point \(x\in X\) with \(g(x)=f(x)\). (Universal maps are thus surjective, and their images have the fixed point property.)NEWLINENEWLINEIn this paper, a continuum \(Y\) has \textit{span zero} if it has \textit{surjective span zero}; i.e., if whenever \(Z\) is a subcontinuum of the square \(Y\times Y\) that misses the standard diagonal, then neither coordinate projection takes \(Z\) onto \(Y\).NEWLINENEWLINEThe main theorem of the paper is that the induced map to the topological suspension of a product of maps from metric continua onto span zero continua is universal. It follows that suspensions and cones over products of span zero continua have the fixed point property. This improves on earlier results involving ``chainable'' instead of (the weaker) ``span zero'', and builds on the result of \textit{M.~M.~Marsh} [Proc. Am. Math. Soc. 132, No. 6, 1849--1853 (2004; Zbl 1047.54026)], that products of maps from metric continua onto span zero continua are universal.