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A Topological Version of Hedetniemi’s Conjecture for Equivariant Spaces

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Abstract

A topological version of the famous Hedetniemi conjecture says: The mapping index of the Cartesian product of two \({\mathbb {Z}}/2\)- spaces is equal to the minimum of their \({\mathbb {Z}}/2\)-indexes. The main purpose of this article is to study the topological version of the Hedetniemi conjecture for G-spaces. Indeed, we show that the topological Hedetniemi conjecture cannot be valid for general pairs of G-spaces. More precisely, we show that this conjecture can possibly survive if the group G is either a cyclic p-group or a generalized quaternion group whose size is a power of 2.

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Notes

  1. A G-equivariant map \(f: X\rightarrow Y\) is a continuous map that also preserves the G-action, i.e., \(f(gx)=gf(x)\) for all \(g\in G\) and \(x\in X\). Moreover, if X and Y are G-simplicial complexes and f is also a simplicial map, then it is called a G-simplicial map.

  2. A G-space Y is called tidy if \({{\,\textrm{ind}\,}}Y ={{\,\mathrm{co-ind}\,}}Y\), where \({{\,\mathrm{co-ind}\,}}Y\) is the maximum k such that there exists a G-equivariant map from \(E_kG\) to Y.

  3. The generalized quaternion group is given by the presentation \(Q_{4n}=\langle a,b: a^n=b^2, a^{2n}=1, b^{-1}ab=a^{-1}\rangle \) where \(n\ge 2\).

  4. It should be noted that the cross-index for G-poset where \(G=\mathbb {Z}_2\) and \({\mathbb {Z}}/p\) were defined respectively in [1, 10].

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Acknowledgements

The authors would like to express their gratitude to the anonymous reviewers for their invaluable feedback and comments on the manuscript, which greatly contributed to enhancing its overall quality.

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Authors and Affiliations

  1. Institut für Informatik, Freie Universität Berlin, Takustraße 9, 14195, Berlin, Germany

    Vuong Bui

  2. LIRMM, Université de Montpellier, 161 Rue Ada, 34095, Montpellier, France

    Vuong Bui

  3. UET, Vietnam National University, Hanoi, 144 Xuan Thuy Street, Hanoi, 100000, Vietnam

    Vuong Bui

  4. Department of Mathematical Sciences, University of Nottingham Ningbo China, 199 Taikang East Road, Ningbo, 315100, China

    Hamid Reza Daneshpajouh

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  1. Vuong Bui
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  2. Hamid Reza Daneshpajouh
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Correspondence to Vuong Bui.

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Bui, V., Daneshpajouh, H.R. A Topological Version of Hedetniemi’s Conjecture for Equivariant Spaces. Combinatorica 44, 441–452 (2024). https://doi.org/10.1007/s00493-023-00079-8

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