Hedetniemi's conjecture from the topological viewpoint

This paper is devoted to studying a topological version of the famous Hedetniemi conjecture which says: The Z/2-index of the Cartesian product of two Z/2-spaces is equal to the minimum of their Z/2-indexes. We fully confirm the version of this conjecture for the homological index via establishing a stronger formula for the homological index of the join of Z/2 -spaces. Moreover, we confirm the original conjecture for the case when one of the factors is an n-sphere. In particular, we answer a question about computing the index of some non-trivial products, raised by Marcin Wrochna. There is a generalization of Hedetniemi's conjecture for hypergraphs by Xuding Zhu. The surprising counterexample of the Hedetniemi conjecture which has been found very recently by Yaroslav Shitov shows that Hedetniemi's conjectu-re, and hence its generalization, only can be valid under additional assumptions. To approach Zhu's conjecture, we establish a topological lower bound for the chromatic number of the categorical product of hypergraphs. Consequently, we enrich the family of known hypergraphs satisfying Zhu'sconjecture. In particular, this leads to a new proof for the fact that Zhu's conjecture is valid for the usual Kneser r- hypergraphs which has been established recently by Hossein Hajiabolhassan and Frederic Meunier as a first non-trivial example of hypergraphs satisfying Zhu's conjecture.(c) 2022 Elsevier Inc. All rights reserved.