Masahiro Takeda
A gauge group is the topological group of automorphisms of a principal bundle. We compute the integral cohomology ring of the classifying spaces of gauge groups of principal U(n)-bundles over the 2-sphere by generalizing the operation for free loop spaces, called the free double suspension.
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Gromov and Ivanov established an analogue of Leray's theorem on cohomology of contractible covers for bounded cohomology of amenable covers. We present an alternative proof of this fact, using classifying spaces of families of subgroups.
In this paper, we compute the rational cohomology groups of the classifying space of a simply connected Kac-Moody group of infinite type. The fundamental principle is "from finite to infinite". That is, for a Kac-Moody group G(A) of infinite type, the input data for computation are the rational cohomology of classifying spaces of parabolic subgroups of G(A)(which are of finite type), and the homomorphisms induced by inclusions of these subgroups. In some special cases, we can further determine the cohomology rings. Our method also applies to study the mod p cohomology of the classifying spaces…
We study the structure of the $E_2$-term of the Rothenberg-Steenrod spectral sequence converging to the mod 3 cohomology of the classifying space of the compact, connected, simply connected, exceptional Lie group of rank 6.
Cohomology of classifying spaces of rank 3 Kac-Moody groups
本文用Weyl群不变量及其商群的直和刻画秩3 Kac-Moody群分类空间的有理数和模p上同调，证明每个素数p都对应p-挠元，并基本确定了有理数上同调的环结构。
We calculate the Weyl group invariants with respect to a maximal torus of the exceptional Lie group $E_6$.
The cohomology of the classifying space of $PU(4)$
本文确定了射影酉群 $PU(4)$ 的分类空间 $BPU(4)$ 的整上同调环及其模 $2$ 上同调的 Steenrod 代数结构。
Nilpotent elements in the cohomology of the classifying space of a connected Lie group
本文研究了连通李群分类空间的模2上同调环中幂零元的存在性，给出了秩最低的紧连通李群的例子。
On the Rothenberg-Steenrod spectral sequence for the mod 3 cohomology of the classifying space of the exceptional Lie group E_8
我们证明了收敛到例外李群E_8的分类空间的模3上同调的Rothenberg-Steenrod谱序列在E_2-水平上不坍缩.
On the Rothenberg-Steenrod spectral sequence for the mod 2 cohomology of classifying spaces of spinor groups
我们计算旋量群spin（n）的模2上同调的余扭积，它是旋量群spin（n）的分类空间的模2上同调的Rothenberg-Steenrod谱序列的E2项.作为这种计算的结果，我们显示了非崩溃的Rothenberg-Steenrod谱序列n > 16。
For a compact Lie group G we show that if the representing spectrum for Borel cohomology generates its category of modules if G is connected. For a closed subgroup H of G we consider the map C^*(BG)--->C^*(BH) and establish the sense in which it is relatively Gorenstein. Throughout, we pay careful attention to the importance of connectedness of the groups.
For n>2, we prove the mod 2 cohomology of the finite Chevalley group Spin_n(F_q) is isomorphic to that of the classifying space of the loop group of the spin group Spin(n).
We show that the cohomology group of the equivariant simplicial de Rham complex is isomorphic to the cohomology group of the classifying space of a semi-direct product group.
This thesis consists of two main parts. In the second part, we recall how a description of local coefficients that Eilenberg introduced in the 1940s leads to spectral sequences for the computation of homology and cohomology with local coefficients. We then show how to construct new equivariant analogues of these spectral sequences for RO(G)-graded Bredon homology and cohomology. Finally, we use these spectral sequences to complete a sample calculation, in which we use the equivariant Serre spectral sequence and the equivariant cohomology of complex projective spaces to compute the cohomology o…
On the mod p cohomology of BPU(p)
研究了射影酉群PU（p）的分类空间的模p上同调.我们首先证明，由于J.F. Adams，Kono和Yagita关于连通紧李群的分类空间的模p上同调的结构在PU（p）情形下成立.最后，证明了当p>3时，射影酉群PU（p）的分类空间由其模p上同调确定为Steenrod代数上的不稳定代数，从而完成了p= 2，3时的工作.
On the Cohomology of the Total Space of Classifying Space for Commutativity in $U(3)$
本文将 $E_{com} U(3)$ 描述为空间图表的同伦余极限，利用相关谱序列计算了其模 $2$ 和模 $3$ 上同调，并给出了有理上同调的环结构。
Let M be a closed simply connected 2n-dimensional manifold. The present paper is concerned with the cohomology of classifying spaces of connected groups of homeomorphisms of M.
On the cohomology and the Chow ring of the classifying space of PGL_p
研究了复射影线性群PGL_p的分类空间的整上同调环和Chow环，当p是奇素数时.特别地，我们完全确定了它的加法结构，并且我们将确定它的乘法结构的问题归结为不变量理论中的问题。
We compute the additive structure of the $\mathrm{RO}(C_n)$-graded Bredon equivariant cohomology of the equivariant classifying space $B_{C_n}\mathrm{SU}(2)$, for any $n$ that is either prime or a product of distinct odd primes, and we also compute its multiplicative structure for $n=2$. In particular, as an algebra over the cohomology of a point, we show that the cohomology of $B_{C_2}\mathrm{SU}(2)$ is generated by two elements subject to a single relation: writing $σ$ for the sign representation of $C_2$ in $\mathrm{RO}(C_2)$, the generators are an element $c$ in dimension $4σ$ and an eleme…
Bounded cohomology of classifying spaces for families of subgroups
本文引入子群族的有界 Bredon 上同调，推广有界上同调，并给出相对可遗性与相对双曲性的上同调刻画。
On the p-primary subgroups of the cohomology of the classifying spaces of PUn
本文扩展了先前的研究，描述了射影酉群 $PU_n$ 的分类空间上同调 $H^s(BPU_n;\mathbb{Z})_{(p)}$ 在 $s<2p+9$ 时的 $p$-准素子群结构，并确定了其在特定维度的同调群性质。
Complementing and extending the Inventiones work of Benson, Grodal, Henke [Group cohomology and control of p-fusion, Invent. Math. 197 (2014), 491--507] we give criteria for a space to have cohomology (strongly) F-isomorphic in the sense of Quillen to the stable elements. We extend results about groups models to fusion systems over discrete $p$-toral groups and profinite groups and provide various applications to the Kan-Thurston Theorem.
On the cohomology of the classifying spaces of $SO(n)$-gauge groups over $S^2$
本文计算了S^2上SO(3)和SO(4)-规范群分类空间的整系数上同调，证明了其同调无挠当且仅当n≤4，并据此分类了n≤4时黎曼曲面上SO(n)-规范群的同伦型。
We compute the cohomology of the unordered configuration spaces of the sphere $S^2$ with integral and with $\mathbb{Z}/p \mathbb{Z}$-coefficients using a cell complex by Fuks, Vainshtein and Napolitano.
Equivariant real cycle class map and Witt-sheaf cohomology of classifying spaces
本文研究等变实循环类映射与Witt-sheaf上同调的关系，通过分类空间的上同调分析，探讨了实代数群的几何分类空间与强实形式之间的联系，并计算了旋群的上同调不变量数量。
Let $\mathbf{B}PU_{n}$ be the classifying space of $PU_n$, the projective unitary group of order $n$, for $n>1$. We use the Serre spectral sequence associated to a fiber sequence $\mathbf{B}U_n\rightarrow\mathbf{B}PU_n\rightarrow K(\mathbb{Z},3)$ to determine the ring structure of $H^{*}(\mathbf{B}PU_{n}; \mathbb{Z})$ up to degree $10$, as well as a family of distinguished elements of $H^{2p+2}(\mathbf{B}PU_{n}; \mathbb{Z})$, for each prime divisor $p$ of $n$. We also study the primitive elements of $H^*(\mathbf{B}U_n;\mathbb{Z})$ as a comodule over $H^*(K(\mathbb{Z},2);\mathbb{Z})$, where the…
For each of the groups $G = O(2), SU(2), U(2)$, we compute the integral and $\mathbb{F}_2$-cohomology rings of $B_\text{com} G$ (the classifying space for commutativity of $G$), the action of the Steenrod algebra on the mod 2 cohomology, the homotopy type of $E_\text{com} G$ (the homotopy fiber of the inclusion $B_\text{com} G \to BG$), and some low-dimensional homotopy groups of $B_\text{com} G$.
A conjecture on homotopy groups of spheres, details on the algebra of higher cohomology operations
二级同调运算理论引出了一个关于一般的高阶上同调运算代数的猜想。这个猜想在这里详细讨论和它的连接与同伦群的领域和亚当斯谱序列的描述。
Supersymmetric field theories and cohomology
这是博士作者的论文。该项目的动机是猜想，霍普金-米勒tmf频谱可以描述的“空间”的共形域论。本文将域论空间构造为以某种类型的域论为对象的范畴的分类空间。如果这样一个范畴具有一个对称的么半群结构，并且它的组成部分形成一个群，那么根据Segal的工作，它的分类空间是一个无限闭路空间，并且定义了一个上同调理论。这已对两类域论进行了研究：（i）对于每个整数n，存在一个范畴SEFT_n，其对象是Stolz-Teichner（1| 1）维n次超欧几里德域论。证明了分类空间|SEFT_n|表示n次K或KO上同调，依赖于域论的系数. (ii)对于每个整数n，存在一个范畴AFT_n，其对象是一类（2| 1）维域论，称为“环形域论”，只使用超几何形式的圆和圆环来定义。证明了分类空间|AFT_n|表示与Tate曲线相关的n次椭圆上同调。据作者所知，这是第一次详细地给出低维超对称域论的定义。
We study the classifying space of a twisted loop group $L_σG$ where $G$ is a compact Lie group and $σ$ is an automorphism of $G$ of finite order modulo inner automorphisms. Equivalently, we study the $σ$-twisted adjoint action of $G$ on itself. We derive a formula for the cohomology ring $H^*(BL_σG)$ and explicitly carry out the calculation for all automorphisms of simple Lie groups. More generally, we derive a formula for the equivariant cohomology of compact Lie group actions with constant rank stabilizers.
Noetherian loop spaces
模p上同调是诺特的闭路空间的类比p-紧群的类大得多（对于p-紧群，模p上同调必须是有限的）。它包含Eilenberg-Mac Lane空间，如无限复射影空间和紧李群的3-连通覆盖。我们研究了这类对象的分类空间BX的上同调，证明了它和预期的一样小，即可以与BCP^ \infty的上同调相比较。我们还证明了BX与单个同伦群中p-紧群的分类空间基本不同。这特别适用于4-连通覆盖的分类空间的李群，并揭示了新的光如何上同调这样一个对象看起来像。
We study the cohomology with twisted coefficients of the geometric realization of a linking system associated to a saturated fusion system $\mathcal{F}$. More precisely, we extend a result due to Broto, Levi and Oliver to twisted coefficients. We generalize the notion of $\mathcal{F}$-stable elements to $\mathcal{F}^c$-stable elements in a setting of cohomology with twisted coefficients by an action of the fundamental group.% or, in other word, with locally constant coefficients. We then study the problem of inducing an idempotent from an $\mathcal{F}$-characteristic $(S,S)$-biset and we show …
Let $G$ be a smooth connected reductive group over a field $k$ and $Γ$ be a central subgroup of $G$. We construct Eilenberg-Moore-type spectral sequences converging to the Hodge and de Rham cohomology of $B(G/Γ)$. As an application, building upon work of Toda and using Totaro's inequality, we show that for all $m\geq 0$ the Hodge and de Rham cohomology algebras of the classifying stacks $B\mathrm{PGL}_{4m+2}$ and $B\mathrm{PSO}_{4m+2}$ over $\mathbb{F}_2$ are isomorphic to the singular $\mathbb{F}_2$-cohomology of the classifying space of the corresponding Lie group. From this we obtain a full…
We present the length, a numerical cohomological index theory, of $ G $-spaces which are cohomology spheres and $ G $ is a $p$-torus or a torus group, where $p$ is a prime. As a consequence, we obtain Borsuk-Ulam and Bourgin-Yang type theorems in this context. A sharper version of the Bourgin-Yang theorem for topological manifolds is also proved. Also, we give some general results regarding the upper and lower bound for the length.
Algebraic models for classifying spaces of fibrations
本文研究了纤维化分类空间$B\operatorname{aut}(X)$的有理同伦型，通过构造幂零dg李代数代数模型，推广了Sullivan-Wilkerson定理并改进了高连通流形自同伦等价的结果。
For any odd prime $p$, we prove that the induced homomorphism from the mod $p$ cohomology of the classifying space of a compact simply-connected simple connected Lie group to the Weyl group invariants of the mod $p$ cohomology of the classifying space of its maximal torus is an epimorphism except for the case $p=3$, $G=E_8$.
Cohomology algebra of orbit spaces of free involutions on the product of projective space and 4-sphere
研究具有射影空间与4-球面乘积模2上同调的有限型空间在自由对合下的商空间上同调代数，并推导等变映射存在性结论。
Bredon cohomology methods in mass partition problems on spheres
本文利用Bredon上同调方法重新证明了关于球面上质量分割问题的定理，展示了拓扑框架在解决此类问题中的灵活性和应用价值。
Motivic cohomology of the Nisnevich classifying space of even Clifford groups
本文计算了分裂偶 Clifford 群的 Nisnevich 分类空间的模 2 动机上同调环，并指出其与 spin 群情形的主要差异在于第二个精细 Stiefel-Whitney 类非平凡。
This paper presents a commutative complex-oriented cohomology theory that realizes the Buchstaber formal group law localized away from 2. Also, the restriction of the classifying map of FB on special unitary cobordism ring localized away from 2 defines a four parameter genus, studied by Hoehn and Totaro.
In this paper we determine the rational homotopy type of the classifying space of a generic Kac-Moody group by computing its rational cohomology ring. As an application we determine the rational homology Hopf algebra of the generic Kac-Moody group.
Finite Chevalley groups and loop groups
设p，e1是不同素数，q是p的幂，G是连通紧李群.证明了存在一个整数b使得有限Chevalley群G（F_q）的分类空间的模上同调同构于Loop群LG的分类空间的模上同调，其中q=p^{ab}，a>0.
Let BP be the p-completed classifying space of a p-group P with p-rank 2. For an odd prime p, by using stable homotopy splitting of BP, we study the decomposition of even dimensional parts of the integral cohomology and the Chow ring of BP.
Cohomology of spaces of Hopf equivariant maps of spheres
本文证明了球面Hopf等变映射空间的有理上同调环与相应Stiefel流形自然同构，并指出其整上同调群的自然映射是满射但非单射。
A note on relative Gelfand-Fuks cohomology of spheres
本文利用有理同伦理论工具研究球面上向量场相对于特殊正交群的 Gelfand-Fuks 上同调，纠正了 Haefliger 的一个断言，并给出 S^3 情形下与保向微分同胚群光滑上同调的一致性及低阶计算。
In this paper we provide a computation of the mod 2 cohomology groups of the third finite subset space of the sphere $S^n$ using known results about the cohomology of the symmetric product of spheres.
Natural operations in Intersection Cohomology
本研究解决了Goresky与MacPherson提出的公开问题，构造出相交上同调的分类单纯集与分类空间，证明其构成无穷loop空间，还研究了深度1分层空间的分类空间性质并提出相关猜想。
Secondary theories for simplicial manifolds and classifying spaces
我们定义了单纯光滑流形的第二理论和特征类，推广了Karoubi的乘法K-理论和光滑流形的乘法上同调群。作为特例，我们得到了单纯光滑流形的Cheeger和Simons的微分特征标群.特殊的例子包括分类空间的李群和李群胚。
Categorifying the concept of topological group, one obtains the notion of a 'topological 2-group'. This in turn allows a theory of 'principal 2-bundles' generalizing the usual theory of principal bundles. It is well-known that under mild conditions on a topological group G and a space M, principal G-bundles over M are classified by either the first Cech cohomology of M with coefficients in G, or the set of homotopy classes [M,BG], where BG is the classifying space of G. Here we review work by Bartels, Jurco, Baas-Bokstedt-Kro, and others generalizing this result to topological 2-groups and eve…
This paper gives the cohomology classification of finitistic spaces X equipped with free actions of the group G = S3 and the orbit space X/G is the integral or mod 2 cohomology quaternion projective space HPn. We have proved that X is the integral or mod 2 cohomology (4n+3)-sphere or the product of 3-sphere and quaternion projective space HPn. Similar results for G = S1 actions are also discussed.
Stable splitting and cohomology of p-local finite groups over the extraspecial p-group of order p^3 and exponent p
设p是一个奇素数。设G是超特殊p-群p_+^{1+2}上的p-局部有限群.本文研究了它们的p-完备分类空间BG的上同调和稳定分裂。
We continue the development of the computability of the second real Johnson-Wilson theory. As ER(2) is not complex orientable, this gives some difficulty even with basic spaces. In this paper we compute the second real Johnson-Wilson theory for products of infinite complex projective spaces and for the classifying spaces for the unitary groups.
The stable cohomology of self-equivalences of connected sums of products of spheres
本文确定了球面乘积连通和的同伦自等价稳定分类空间的上同调，并将其用李图复形同调表达。
We compute the Chow-Witt rings of the classifying spaces for the symplectic and special linear groups. In the structural description we give, contributions from real and complex realization are clearly visible. In particular, the computation of cohomology with $\mathbf{I}^j$-coefficients is done closely along the lines of Brown's computation of integral cohomology for special orthogonal groups. The computations for the symplectic groups show that Chow-Witt groups are a symplectically oriented ring cohomology theory. Using our computations for special linear groups, we also discuss the question…
Special generic maps are higher dimensional versions of Morse functions with exactly two singular points, characterizing spheres topologically except $4$-dimensional cases: in these cases standard spheres are characterized. Canonical projections of unit spheres are special generic. In suitable cases, it is easy to construct special generic maps on manifolds represented as connected sums of products of spheres for example. It is an interesting fact that these maps restrict the topologies and the differentiable structures admitting them strictly in various cases. For example, exotic spheres, whi…
This paper generalizes the concept of index and co-index and some related results for free actions of G = S0 on a paracompact Hausdorff space which were introduced by Conner and Floyd. We define the index and co-index of a finitistic free G-space X, where G = Sd , d = 1 or 3 and prove that the index of X is not more than the mod 2 cohomology index of X. We observe that the index and co-index of a (2n + 1)-sphere (resp. (4n+3)-sphere) for the action of componentwise multiplication of G = S1 (resp. S3) is n. We also determine the orbit spaces of free actions of G = S3 on a finitistic space X wit…
We consider certain rational homotopical conditions of simly connected CW complex $X$ such that the rational cohomology of the classifying space $Baut_1X$ for fibrations with two-stage fibre $X$ is (not) free. First, we consider when is $Baut_1X$ a rational factor of $Baut_1(X\times S^n)$ for an odd-integer $n$ and observe for a non-formal elliptic space $X$ of rank 3. Second, we compute the Sullivan minimal models of $Baut_1X$ when $X$ are certain non-formal pure spaces of rank 5.
Let $G$ be a connected Lie group of rank one. In this paper the existence of free actions of group $G$ on spheres, real projective spaces and lens spaces has been studied. Most of the results have been obtained for finitistic spaces with cohomology ring isomorphic to the cohomology ring of these spaces. The cohomology algebra of the orbit space has been determined in the case of free $G$-action. The non-existence of $G$-equivariant maps from spheres to cohomology spheres has also been established.
Free Circle Actions on the Product of Three Spheres
本文将三个球面乘积上自由S^0-作用的轨道空间分类推广到自由S^1-作用，并建立Borsuk-Ulam型定理。
As Reeb's theorem shows, Morse functions with exactly two singular points on closed manifolds are very simple and important. They characterize spheres whose dimensions are not $4$ topologically and the $4$-dimensional unit sphere. Special generic maps are generalized versions of these maps. Canonical projections of unit spheres are special generic. Studies of Saeki and Sakuma since the 1990s, followed by Nishioka and Wrazidlo, show that the differentiable structures of the spheres and the homology groups of the manifolds (in several classes) are restricted. We see special generic maps are attr…
The equivariant cohomology for actions of compact connected abelian groups and elementary abelian p-groups have been widely studied in the last decades. We study some of these results on actions of finite cyclic groups over a field of positive characteristic. In particular, we study the module structure of the equivariant cohomology over the cohomology of the classifying space and we provide a criterion for freeness over this ring and the polynomial subring of it. We apply these results to canonical actions of finite cyclic groups on polyhedral products arising from the boundary of a polygo
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