Weak generalized topological Hedetniemi conjecture

The conjecture predicts that two finite free spaces whose individual mapping indices are small should have a product with similarly controlled index. Bui and Daneshpajouh established a broad index-one counterexample in 2023, and a newer preprint claims counterexamples at every prescribed index.

Known results

• For $G$ neither a cyclic $p$-group nor a generalized quaternion group, Bui and Daneshpajouh constructed finite free $G$-complexes with $\operatorname{ind}K_1=\operatorname{ind}K_2=1$ but $\operatorname{ind}(K_1\times K_2)=0$ (2023).
• For cyclic $p$-groups and generalized quaternion groups of $2$-power order, they proved the corresponding index-one implication.
• They formulated the higher-index statements $HCT_n(G)$ and $HCX_n(G)$ and proved $HCX_n(G)\Rightarrow HCT_n(G)$.

August 25, 2026 higher-index counterexamples

The preprint On a Weak Form of the Topological Hedetniemi Conjecture claims that the earlier counterexamples extend from index-one factors to factors of every prescribed index $n$, yielding arbitrarily strong failure for groups outside the cyclic prime-power and generalized quaternion cases. This is a claimed resolution of the stated generalized problem, not an independently verified result, and it does not address the original $\mathbb{Z}/2$ conjecture.

Current status (as of August 2026): The index-one generalized conjecture is known to fail for the specified groups, and a preprint claims failure at every prescribed index $n$; that higher-index claim remains unverified, while the original $\mathbb{Z}/2$ problem is separate.