AI 中文总结
针对推广拓扑Hedetniemi猜想在非素数幂阶循环群、非广义四元数群G下不成立的结论，本文证明反例中复形与其乘积的拓扑指数差值可任意大，即该猜想弱形式对这类群最强意义下不成立。
AI 中文摘要
拓扑Hedetniemi猜想断言，两个Z/2-复形乘积的拓扑指数等于它们各自指数的最小值。Bui和Daneshpajouh研究了该猜想向G-空间的自然推广，并证明当群G既不是素数幂阶循环群也不是广义四元数群时，这一推广猜想不成立。更确切地说，对于任意此类群G，他们构造了有限自由G-单纯复形，每个的拓扑指数均为1，而它们的乘积的拓扑指数为0。\n本注记的目的是证明这种差异可以任意大。实际上，对于每个整数n≥1，我们证明只要G既不是素数幂阶循环群也不是广义四元数群，就存在两个有限自由G-单纯复形，每个的拓扑指数均为n，而它们的乘积的拓扑指数为0。特别地，对于这些群而言，对应的推广拓扑Hedetniemi猜想的弱形式在最强的意义下不成立。
英文摘要
The topological Hedetniemi conjecture asserts that the topological index of the product of two \(\mathbb{Z}/2\)-complexes is equal to the minimum of their respective indices. Bui and Daneshpajouh studied a natural generalization of this conjecture to \(G\)-spaces and proved that this generalized conjecture fails whenever the group \(G\) is neither cyclic of prime-power order nor generalized quaternion. More precisely, for any such group \(G\), they constructed finite free \(G\)-simplicial complexes, each having topological index one, whose product has topological index zero. The purpose of this note is to demonstrate that this discrepancy can be made arbitrarily large. Indeed, for every integer \(n\geq 1\), we show that there exist two finite free \(G\)-simplicial complexes, each of topological index $n$, whose product has topological index zero, provided that \(G\) is neither cyclic of prime-power order nor generalized quaternion. In particular, the corresponding weak form of the generalized topological Hedetniemi conjecture fails in the strongest possible sense for those groups.
