Dyer-Lashof代数中的Adem关系与模不变量

Homology operations revisited

The mod p homology of E-infinity spaces is a classical topic in algebraic topology traditionally approached in terms of Dyer--Lashof operations. In this paper, we offer a new perspective on the subject by providing a detailed investigation of an alternative family of homology operations equivalent to, but distinct from, the Dyer--Lashof operations. Among other things, we will relate these operations to the Dyer--Lashof operations, describe the algebra generated by them, and use them to describe the homology of free E-infinity spaces. We will also investigate the relationship between the operat…

Algebraic infinite delooping and derived destabilization

Working over the prime field of characteristic two, consequences of the Koszul duality between the Steenrod algebra and the big Dyer-Lashof algebra are studied, with an emphasis on the interplay between instability for the Steenrod algebra action and that for the Dyer-Lashof operations. The central algebraic framework is the category of length-graded modules over the Steenrod algebra equipped with an unstable action of the Dyer-Lashof algebra, with compatibility via the Nishida relations. A first ingredient is a functor defined on modules over the Steenrod algebra that arose in the work of Kuh…

An algorithmic search for $\mathcal{A}$-annihilated classes in the Dyer-Lashof algebra and $H_*QS^0$ I. Closed form for low lengths and tables in low dimensions

The aim of this work is to publicise some computational results involving tables which contain $\mathcal{A}$-annihilated monomials, excluding square classes, in the Dyer-Lashof algebra and $H_*QS^0$; our computations go up to dimension $1.1\times 10^7$ but the tables in this paper only announce results up to dimension $2^{17}=131072$ and full tables would be available upon request. The theoretical background for our computations is provided by work of Curtis \cite{Curtis} and Wellington \cite{Wellington} on the $\mathcal{A}$-module structure of the Dyer-Lashof algebra as well as $H_*QS^0$. It …

On the relation between Dyer-Lashof algebra and the hit problems

The aim of this note is to use geometric methods to study the hit problem of Peterson for $H_*\mathbb{R} P^{\times k}$ as well as the symmetric hit problem of Janfada and Wood for $H_*BO(k)$. We continue by exploring the applications of the results of \cite{Zare-symmetric} on the $\mathcal{A}$-annihilated generators of $H_*QX$ to obtain a family of generic `new' examples of $\mathcal{A}$-annihilated in $H_*(\mathbb{Z}\times BO)$ and $H_*BO$, i.e. the case of stable symmetric hit problem, where an essential step is provided by the infinite loop space structure on $BO$ implied by the Bott period…

配边与同调中的西田关系（扩展摘要）

Nishida Relations in Bordism and Homology (extended abstract)

本文研究非定向配边中 Dyer-Lashof 运算与 Landweber-Novikov 运算之间的西田关系，并利用这些关系通过同调示性数表示覆盖流形代数 $N_*Σ$，证明基于覆盖空间运算及 D-环和 Q-环的性质。

同调运算与余单迭代闭路空间

Homology operations and cosimplicial iterated loop spaces

若X是上单$E_{n+1}$空间，则Tot（X）是$E_{n+1}$空间，且其模2同调$H_*(Tot(X))$具有Dyer-Lashof和Browder运算.人们自然会问，收敛到$H_*(Tot(X))$的谱序列是否允许兼容运算。本文对此问题给予肯定的回答。

Modular coinvariants and the mod p homology of QS^k

We use modular invariant theory to establish a complete set of relations of the mod $p$ homology of $\{QS^k\}_{k\geq0}$, for $p$ odd, as a ring object in the category of coalgebras (also known as a coalgebraic ring or a Hopf ring). We also describe the action of the mod $p$ Dyer-Lashof algebra as well as the mod $p$ Steenrod algebra on the coalgebraic ring.

同伦论中 $δ$-代数与棱柱的札记

Notes on $δ$-algebras and prisms in homotopy theory

本文围绕 $p$-进 $K$-理论中的棱柱结构，梳理其向结构化 Lubin-Tate 谱 Frobenius 提升的推广，并探讨借助 $L$-典型棱柱延伸至 $K(n)$ 局部邻域的可能。

高度 n 处 K(n-1)-局部化 Morava E-理论上的幂运算

Power Operations on $K(n-1)$-Localized Morava $E$-theory at Height $n$

本文计算了对称群的 K(n-1)-局部化 E_n 理论，给出全幂运算 ψ^p_F 的模解释及 Dyer-Lashof 代数结构，并在 n=2 时导出显式公式，揭示其与椭圆曲线、模形式和 p-可除群的联系。