论文标题
产生功能和拓扑复杂性
Generating functions and topological complexity
论文作者
论文摘要
我们研究了理性的猜想，该猜想指出（a）正式功率系列$ \ sum_ {r \ ge 1} \ tc_ {r+1}（x）（x）\ cdot x^r $代表$ x $的合理函数，单个订单2 pole 2 in $ x = 1 $ and（b）pole pole cat $ x等等$ \ cat $ \ cat（x）。这里$ x $是有限的CW-Complex，对于$ r \ ge 2 $，符号$ \ tc_r（x）$表示其$ r $ $ th的顺序拓扑复杂性。我们分析了一个示例（违反了Ganea的猜想），并得出结论，理性猜想的一部分通常是错误的。此外，我们建立了合理性猜想的共同学版本。
We examine the rationality conjecture which states that (a) the formal power series $\sum_{r\ge 1} \tc_{r+1}(X)\cdot x^r$ represents a rational function of $x$ with a single pole of order 2 at $x=1$ and (b) the leading coefficient of the pole equals $\cat(X)$. Here $X$ is a finite CW-complex and for $r\ge 2$ the symbol $\tc_r(X)$ denotes its $r$-th sequential topological complexity. We analyse an example (violating the Ganea conjecture) and conclude that part (b) of the rationality conjecture is false in general. Besides, we establish a cohomological version of the rationality conjecture.
