AI 中文总结
本文通过Behrens-Kuhn的Whitehead猜想结果，建立了圆上恒等函子Goodwillie塔的p-局部等价，将其逼近识别为无穷环空间，并引入Goodwillie演算的Gray序列类比以简化同伦群计算。
AI 中文摘要
我们研究了在圆上求值的恒等函子的Goodwillie塔。我们的主要结果是从Behrens和Kuhn关于Whitehead猜想的工作中得到的p-局部等价 $\Omega P_{p^k}I(S^1) \simeq \mathbb{Z} \times \Omega^{\infty+2k}\tau_{>0}\mathbb{Sp}^{p^k}$。特别地，这确定了圆的Goodwillie逼近为无穷环空间，这一现象在一般情况下并不成立。该等价还将它们的同伦群的计算归结为一个稳定问题，使其可通过诸如Adams谱序列和计算机计算等工具来处理。我们将这些计算置于一个更一般的框架中，通过发展Gray序列的Goodwillie演算类比，该序列关联了塔的连续各层的同伦群。
英文摘要
We study the Goodwillie tower of the identity evaluated at the circle. Our main result is the p-local equivalence $ΩP_{p^k}I(S^1) \simeq \mathbb{Z} \times Ω^{\infty+2k}τ_{>0}\mathbb{Sp}^{p^k}$ obtained from the work of Behrens and Kuhn on the Whitehead conjecture. In particular, this identifies the Goodwillie approximations of the circle as infinite loop spaces, a phenomenon which does not hold in general. The equivalence also reduces the computation of their homotopy groups to a stable problem, making them accessible to tools such as the Adams spectral sequence and to computer calculations. We place these computations in a more general framework by developing a Goodwillie-calculus analogue of the Gray sequence, which relates the homotopy groups of successive stages of the tower.
