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Universality of maps on suspensions over products of span zero continua. (English) Zbl 1282.54025

A continuum is a connected compact Hausdorff space; maps are always assumed to be continuous. A map \(f:X\to Y\) between continua is 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.)
In this paper, a continuum \(Y\) has span zero if it has 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 \(Y\).
The 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 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.

MSC:

54F15 Continua and generalizations
54B10 Product spaces in general topology
54C10 Special maps on topological spaces (open, closed, perfect, etc.)
54H25 Fixed-point and coincidence theorems (topological aspects)

Citations:

Zbl 1047.54026
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