Nobuaki Yagita
In this note we construct approximations by smooth projective varieties of some Eienberg-MacLane spaces in the $A^1$-homotopy category. Using these, we study the cycle maps from Chow rings to etale cohomology rings.
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The paper gives an answer to the question whether the Galois, separable or etale extension of an Eilenberg-MacLane spectrum in the category of ring spectra is again an Eilenberg-MacLane spectrum. We get a full answer in the Galois case. The question in separable and etale cases is answered only under additional hypothesis.
We compute the mod-2 connected KU-homology of the Eilenberg-MacLane space K(Z/2,2), using a novel Adams spectral sequence analysis.
Let $Q$ denote the cyclic group of order two. Using the Tate diagram we compute the $RO(Q)$-graded coefficients of Eilenberg-MacLane $Q$-spectra and describe their structure as a module over the coefficients of the Eilenberg-MacLane spectrum of the Burnside Mackey functor. If the underlying Mackey functor is a Green functor, we also infer the multiplicative structure on the $RO(Q)$-graded coefficients.
K. Borsuk in 1979, in the Topological Conference in Moscow, introduced the concept of the capacity of a compactum and asked some questions concerning properties of the capacity of compacta. In this paper, we give partial positive answers to three of these questions in some cases. In fact, by describing spaces homotopy dominated by Moore and Eilenberg-MacLane spaces, we obtain the capacity of a Moore space $M(A,n)$ and an Eilenberg-MacLane space $K(G,n)$. Also, we compute the capacity of the wedge sum of finitely many Moore spaces of different degrees and the capacity of the product of finitely…
We compute ku^*(K(Z/p,2)) and ku_*(K(Z/p,2)), the connective KU-cohomology and connective KU-homology groups of the mod-p Eilenberg-MacLane space K(Z/p,2), using the Adams spectral sequence. We obtain a striking interaction between h_0-extensions and exotic extensions. The mod-p connective KU-cohomology groups, computed elsewhere, are needed in order to establish higher differentials and exotic extensions in the integral groups.
We note that, for any natural $k$ and every natural $l$ between $k$ and $2k$, there exists a group $π$ with $\cat K(π,1)=k$ and $\TC(K(π,1))=l$. Because of this, we can set up a problem of searching of purely group-theoretical description of $\TC(K(π,1))$ as an invariant of $π$.
The connective Morava K-theory of the second mod p Eilenberg-MacLane space
本文开发了计算空间连通第n阶Morava K-理论的工具，利用全系系数定理实现同调与上同调版本的镜像计算，并具体计算了第二个模p Eilenberg-MacLane空间的该理论。
We filter the equivariant Eilenberg Maclane spectrum $H\underline{\mathbb{F}}_p$ using the mod $p$ symmetric powers of the equivariant sphere spectrum, $\mathrm{Sp}_{\mathbb{Z}/p}^{\infty}(Σ^{\infty G}S^0)$. When $G$ is a $p$-group, we show that the layers in the filtration are the Steinberg summands of the equivariant classifying spaces of $(\mathbb{Z}/p)^n$ for $n=0, 1, 2, \ldots$. We show that the layers of the filtration split after smashing with $H\underline{\mathbb{F}}_p$. Along the way, we produced a general computation of the geometric fixed points of $H\underline{\mathbb{Z}}$ and $H\u…
The connective KO theory of the Eilenberg-MacLane space K(Z/2,2)
本研究借助Adams谱序列算出K(Z/2,2)的连通KO同调与上同调，依托既有复ku群研究发现二者对偶关系，还得到Spin流形Stiefel-Whitney类的新结论。
We prove that the $G$-equivariant mod $p$ Eilenberg--MacLane spectrum arises as an equivariant Thom spectrum for any finite, $p$-power cyclic group $G$, generalizing a result of Behrens and the second author in the case of the group $C_2$. We also establish a construction of $\mathrm{H}\underline{\mathbb{Z}}_{(p)}$, and prove intermediate results that may be of independent interest. Highlights include constraints on the Hurewicz images of equivariant spectra that admit norms, and an analysis of the extent to which the non-equivariant $\mathrm{H}\mathbb{F}_p$ arises as the Thom spectrum of a mo…
The connective KO-theory of the Eilenberg-MacLane space K(Z_2,2), I: the E_2 page
本研究计算了收敛于第二模2 Eilenberg-MacLane空间连通KO理论的Adams谱序列E_2页，分析了对应上同调作为Steenrod代数子代数模的结构，完整谱序列分析将后续发表。
Primary cohomology operations, i.e., elements of the Steenrod algebra, are given by homotopy classes of maps between Eilenberg--MacLane spectra. Such maps (before taking homotopy classes) form the topological version of the Steenrod algebra. Composition of such maps is strictly linear in one variable and linear up to coherent homotopy in the other variable. To describe this structure, we introduce a hierarchy of higher distributivity laws, and prove that the topological Steenrod algebra satisfies all of them. We show that the higher distributivity laws are homotopy invariant in a suitable sens…
For a connected CW-complex, we let $SNT(X)$ be the set of all homotopy types $[Y]$ such that the Postnikov approximations $X^{(n)}$ and $Y^{(n)}$ of $X$ and $Y$, respectively, are homotopy equivalent for all positive integers $n$. In 1992, McGibbon and Møller (\cite[page 287]{MM}) raised the following question: Is $SNT(Σ\mathbb C P^\infty) = *$ or not? In this article, we give an answer to the more generalized version of this query: The set of all the same $n$-types of the suspended wedge sum of the Eilenberg-MacLane spaces of various types of both even and odd integers is the set which consis…
We prove the uniqueness of twisted K-theory in both the real and complex cases using the computation of the K-theories of Eilenberg-MacLane spaces due to Anderson and Hodgkin. As an application of our method, we give some vanishing results for actions of Eilenberg-MacLane spaces on K-theory spectra.
The $H \underline{\mathbb{F}}_2$-homology of $C_2$-equivariant Eilenberg-MacLane spaces
本文将 Ravenel-Wilson Hopf 环技术扩展至 $C_2$-等变同伦理论，计算了与常值 Mackey 函子关联的 $C_2$-等变 Eilenberg-MacLane 空间的 $RO(C_2)$-分次同调，并探讨了相关的扭曲 bar 谱序列。
Eilenberg-MacLane spaces and stabilisation in homotopy type theory
本文在同伦类型论中研究空间与映射的解环，证明部分空间和群同态具有唯一解环，并讨论其在 Eilenberg-MacLane 空间与上同调中的应用。
On Topological André-Quillen homology of Eilenberg-MacLane spectra
本文基于Dundas等人工作，计算了Eilenberg-MacLane谱（如Hℤ和Hℤ/p^n）的拓扑André-Quillen同调，方法类似Cartan计算Steenrod代数，且更简短。
On formality of diagrams of Eilenberg-MacLane spaces
本文证明了任意高度n≥1的Eilenberg-MacLane空间图式在有理数上形式，从而其谱序列在第二页坍缩。通过函子计算论证了在不包含有理数的固定交换环上形式性不成立。
A complete classification of the rational realization of gap-two products of Eilenberg--MacLane spaces as classifying spaces
本研究针对单连通有理同调有限型的$\pi$有限空间，探讨Eilenberg--MacLane空间的间隙二乘积作为分类空间的有理实现，证明其有理同伦等价的充要条件，并确定其极小Sullivan模型的同构类及有理同伦类型的唯一性情况。
Eilenberg-MacLane spaces in A-homotopy theory
本文在A-同伦理论中构造了具有任意指定同伦群的图，证明了相对A-同伦群与经典相对同伦群的等同，并建立了离散Blakers-Massey定理及Postnikov系统和Whitehead塔。
In this short note we study the topological Hoschschild homology of Eilenberg-MacLane spectra for finite cyclic groups. In particular, we show that the Eilenberg-MacLane spectrum H(Z/p^k) is a Thom spectrum for any prime p (except, possibly, when p=k=2) and we also compute its topological Hoschshild homology. This yields a short proof of the results obtained by Brun, and by Pirashvili except for the anomalous case p=k=2.
The Eilenberg-MacLane Spectrum of \mathbb{F}_1
本文通过Segal解环函子计算F1模型的n重解环，得到n维球面及其自由部分交换幺半群结构，等价于具有自由部分对称幺半结构的自由部分严格n-范畴的神经。
On The Rational Realization of Even-dimensional Spheres and Products of Eilenberg--MacLane Spaces as Classifying Spaces
本文研究分类空间的有理实现问题，利用Sullivan极小模型导子李代数中Gottlieb元素的结构结果，证明偶维球面及特定Eilenberg-MacLane空间乘积无法作为π-有限空间的第一类自同态分类空间，给出K(Q^r,n)实现为分类空间时X的有理同伦型。
We prove that for a finitely generated group $G$, the second stable homotopy group $π_2^S(K(G,1))$ of the Eilenberg-Maclane space $K(G,1)$ is completely determined by the Schur multiplier $H_2(G)$. We also prove that the second stable homotopy group $π_2^S(K(G,1))$ is equal to the Schur multiplier $H_2(G)$ for a torsion group $G$ with no elements of order $2$ and show that for such groups, $π_2^S(K(G,1))$ is a direct factor of $π_{3}(SK(G,1))$, where $S$ denotes suspension and $π_2^S$ the second stable homotopy group. We compute $π_{3}(SK(G,1))$ and $π_2^S(K(G,1))$ for symmetric, alternating, …
In an earlier paper of Cadek, Vokrinek, Wagner, and the present authors, we investigated an algorithmic problem in computational algebraic topology, namely, the computation of all possible homotopy classes of maps between two topological spaces, under suitable restriction on the spaces. We aim at showing that, if the dimensions of the considered spaces are bounded by a constant, then the computations can be done in polynomial time. In this paper we make a significant technical step towards this goal: we show that the Eilenberg-MacLane space K(Z,1), represented as a simplicial group, can be equ…
We are familiar with properties and structure of topological spaces. One of the powerful tools, which help us to figure out the structure of topological spaces is (Leray- Serre) spectral sequence. Although Eilenberg-Maclane space plays important roles in topology, and respectively geometry. Actually finding cohomology groups of this space can be useful for classifying spaces, and also homotopy groups structure of these groups. This paper discusses how to compute cohomology groups of Eilenberg-Maclane spaces $K(\mathbb{Z}, 4)$ and $K(\mathbb{Z}, 5)$ (cohomology degree less than $11$). Furthermo…
In this paper, we calculate the $RO(G)$-graded coefficients of $H\underline{\mathbb{Z}}$, the Eilenberg-MacLane spectrum of constant Mackey functor for quaternion group $Q_8$.
Geometric Constructions of Mod $p$ Cohomology Operations
本文基于Brown可表示性和Almgren同构定理，构造了从球面循环空间到圆盘相对循环空间的显式映射，用以表示模$p$上同调中的所有Steenrod平方、Steenrod幂和Bockstein同态。
This paper determines the RO(G)-graded Eilenberg-MacLane cohomology of the real, infinite, equivariant Grassmannians in the case G=Z/2. Possible connections with motivic characteristic classes for quadratic bundles are briefly discussed.
We prove that the $C_2$-equivariant Eilenberg-MacLane spectrum associated to the constant Mackey functor $\underline{\mathbb{F}}_2$ is equivalent to a Thom spectrum over ${Ω^ρS^{ρ+ 1}}$.
Stability Patterns for Spherical and Projective Braid Groups
本文利用Cohen–Pakianathan的成果与$\text{FI}$同调理论，将稳定性模式推广到球面$S^2$和射影平面$\rp^2$上的辫群，证明其等变同调分别具有表示稳定性与稳定周期性。
We give a particular choice of the higher Eilenberg-MacLane maps by a recursive formula.This choice leads to a simple description of the homotopy operations for simplicial Z/2-algebras.
We ask whether a morphism $g$ in a triangulated category $C$ endowed with a weight structure "kills weights" (between an integer $m$ and some $n\ge m$). If $g=id_M$ (where $M\in Obj C$) and $C$ is Karoubian, then $g$ kills weights $m,\dots,n$ whenever there exists a weight decomposition of $M$ that "avoids" these weights (in the sense earlier defined by Wildeshaus). We prove the equivalence of several definitions for killing weights. In particular, we describe a family of cohomological functors that "detects" this notion. We also prove that $M$ is without weights $m,\dots, n$ (i.e., a decompos…
The paper is devoted to study the space of multiplicative maps from the Eilenberg-MacLane spectrum $H\Z$ to an arbitrary ring spectrum $R$. We try to generalize the approach of Schwede from "Formal groups and stable homotopy of commutative rings", where the special case of the mentioned above problem was solved in full generality. Among other results we propose a definition of a formal group law in any ring spectrum, which might be of interest in its own.
The well-known theorem of Eilenberg and Ganea expresses the Lusternik - Schnirelmann category of an aspherical space as the cohomological dimension of its fundamental group. In this paper we study a similar problem of determining algebraically the topological complexity of the Eilenberg-MacLane spaces. One of our main results states that in the case when the fundamental group is hyperbolic in the sense of Gromov the topological complexity of an aspherical space $K(π, 1)$ either equals or is by one larger than the cohomological dimension of $π\times π$. We approach the problem by studying essen…
In this survey paper on commutative ring spectra we present some basic features of commutative ring spectra and discuss model category structures. As a first interesting class of examples of such ring spectra we focus on (commutative) algebra spectra over commutative Eilenberg-MacLane ring spectra. We present two constructions that yield commutative ring spectra: Thom spectra associated to infinite loop maps and Segal's construction starting with bipermutative categories. We define topological Hochschild homology, some of its variants, and topological Andre-Quillen homology. Obstruction theory…
Some asymptotic formulae for torsion in homotopy groups
受有理同伦群秩渐近估计及局部双曲性研究启发，本文证明了同伦群挠秩的两个渐近公式，并据此得到了多种空间在悬垂后同伦群挠秩的显式定量渐近下界。
We show that in the category of complex algebraic varieties, the Eilenberg--Moore spectral sequence can be endowed with a weight filtration. This implies that it degenerates if all involved spaces have pure cohomology. As application, we compute the rational cohomology of an algebraic $G$-variety $X$ ($G$ being a connected algebraic group) in terms of its equivariant cohomology provided that $H_G(X)$ is pure. This is the case, for example, if $X$ is smooth and has only finitely many orbits. We work in the category of mixed sheaves; therefore our results apply equally to (equivariant) intersect…
Homotopy type theory is a version of Martin-Löf type theory taking advantage of its homotopical models. In particular, we can use and construct objects of homotopy theory and reason about them using higher inductive types. In this article, we construct the real projective spaces, key players in homotopy theory, as certain higher inductive types in homotopy type theory. The classical definition of RP(n), as the quotient space identifying antipodal points of the n-sphere, does not translate directly to homotopy type theory. Instead, we define RP(n) by induction on n simultaneously with its tauto…
Algebraic cycles on complex projective space P(V) are known to have beautiful and surprising properties. Therefore, when V carries a real or quaternionic structure, it is natural to ask for the properties of the groups of real or quaternionic algebraic cycles on P(V). In this paper and its sequel the homotopy structure of these cycle groups is determined. They bear a direct relationship to characteristic classes for the classical groups, and functors in K-theory extend directly to these groups. These groups give rise to E-infinity-ring spaces, and the maps extending the K-theory functors are r…
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