The main result of this paper is that for n = 3,4,5 and k = n-2, every Borsuk continuous set-valued map of the closed ball in the n-dimensional Euclidean space with values which are one-point sets or sets homeomorphic to the k-sphere has a fixed point. Our approach fails for (k,n) = (1,4). A relevant counterexample (for the homological method, not for the fixed point conjecture) is indicated.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm175-1-4,
author = {Dariusz Miklaszewski},
title = {A fixed point conjecture for Borsuk continuous set-valued mappings},
journal = {Fundamenta Mathematicae},
volume = {173},
year = {2002},
pages = {69-78},
zbl = {1035.54012},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm175-1-4}
}
Dariusz Miklaszewski. A fixed point conjecture for Borsuk continuous set-valued mappings. Fundamenta Mathematicae, Tome 173 (2002) pp. 69-78. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm175-1-4/