$E_n$ ring spectra and Dyer-Lashof operations

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。

Power operations and coactions in highly commutative homology theories

Power operations in the homology of infinite loop spaces, and $H_\infty$ or $E_\infty$ ring spectra have a long history in Algebraic Topology. In the case of ordinary mod p homology for a prime p, the power operations of Kudo, Araki, Dyer and Lashof interact with Steenrod operations via the Nishida relations, but for many purposes this leads to complicated calculations once iterated applications of these functions are required.On the other hand, the homology coaction turns out to provide tractable formulae better suited to exploiting multiplicative structure. We show how to derive suitable for…

Homology operations and cosimplicial iterated loop spaces

If X is a cosimplical $E_{n+1}$ space then Tot(X) is an $E_{n+1}$ space and its mod 2 homology $H_*(Tot(X))$ has Dyer-Lashof and Browder operations. It's natural to ask if the spectral sequence converging to $H_*(Tot(X))$ admits compatible operations. In this paper I give a positive answer to this question.

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。

广义串拓扑运算

Generalized string topology operations

我们证明了Chas-Sullivan圈积（纤维上的Pontrjagin积和基上的交积的组合）对闭定向流形M上的纤维幺半群E的全空间同调是有意义的.更一般地，托姆谱E^{-TM}是环形谱。类似地，E上的纤维模定义了E^{-TM}上的模。纤维幺半群包含主丛的伴随丛，并且其构造对于主丛的映射是自然的。这种自然性意味着H_*（LM）上由圈积产生的代数结构的同伦不变性。若M=BG是紧李群的无限维分类空间，则得到一个定义良好的前环谱，定义为BG的弦拓扑.若E具有小n方体算符的纤维作用量，则E^{-TM}是E_n-环谱.这给出了结合纤维上的Dyer-Lashof操作和基底上的Steenrod操作的同源操作。我们给出了几个例子，新的操作给同调的洞察力，从纽结理论，复杂的几何，规范理论，同伦理论借用。