We develop a theory of generalized Hopf invariants in the setting of sectional category. In particular, we show how Hopf invariants for a product of fibrations can be identified as shuffle joins of Hopf invariants for the factors. Our results are applied to the study of Farber’s topological complexity for two-cell complexes, as well as to the construction of a counterexample to the analogue for topological complexity of Ganea’s conjecture on Lusternik–Schnirelmann category.

中文翻译：

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截面类别的Hopf不变量及其在拓扑机器人中的应用

我们在截面类别的设置中发展了广义Hopf不变量的理论。特别是，我们展示了如何将纤维化产物的Hopf不变量确定为该因子的Hopf不变量的混洗连接。我们的研究结果用于研究两细胞复合物的Farber拓扑复杂性，以及构造Ganea关于Lusternik–Schnirelmann类的猜想的拓扑复杂性类似物的反例。