The Structure of RO(G)-Graded Homotopy of Eilenberg-MacLane Spectra for Cyclic Two-Groups and the Slice Spectral Sequences

We study the RO(G)-graded homotopy Mackey functors of Eilenberg-Mac Lane spectra for cyclic p-groups. One innovation is the use of the generalized Tate squares introduced by Greenlees-May in the computations. We exploit the power of these generalized Tate squares further by applying them to the study of the equivariant slice spectral sequence invented by Dugger which is later generalized by Hill-Hopkins-Ravenel in their solution of the Kervaire invariant problem. The Tate squares for different families provide stratification of the slice spectral sequences. We deduce vanishing lines and transchromatic phenomenon in the negative cones of these spectral sequences, extending the work of Meier-Shi-Zeng on the positive cones. We also compute RO(G)-graded coefficients in some other cases, as illustrations of the usefulness of the Tate squares in equivariant computations, especially when dealing with the multiplicative structures.我们针对循环p群，研究艾伦伯格-麦克莱恩谱（Eilenberg-Mac Lane spectra）的RO(G)-分次同伦Mackey函子。本研究的一项创新在于，计算过程中采用了格林利斯-梅（Greenlees-May）提出的广义泰特平方（generalized Tate squares）。我们进一步拓展了这类广义泰特平方的应用场景，将其用于达格（Dugger）提出、后经希尔-霍普金斯-拉文内尔（Hill-Hopkins-Ravenel）在解决凯尔维尔不变量问题时推广的等变切片谱序列（equivariant slice spectral sequence）的相关研究。针对不同族类的泰特平方，可实现切片谱序列的分层。我们推导得到了这类谱序列负锥区域内的消失线与跨色现象，拓展了迈耶-施-曾（Meier-Shi-Zeng）针对正锥区域的既有研究工作。此外，我们还在若干其他场景下计算了RO(G)-分次系数，以此展示泰特平方在等变计算——尤其是涉及乘法结构（multiplicative structures）的等变计算——中的实用价值。