Cohomology of the classifying spaces of $U(n)$-gauge groups over the 2-sphere

连通李群的分类空间余弦学中的幂零元

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

We show that the Rothenberg--Steenrod spectral sequence converging to the mod 3 cohomology of the classifying space of the exceptional Lie group E_8 does not collapse at the E_2-level.

On the Rothenberg-Steenrod spectral sequence for the mod 2 cohomology of classifying spaces of spinor groups

We compute the cotorsion product of the mod 2 cohomology of spinor group spin(n), which is the E_2-term of the Rothenberg-Steenrod spectral sequence for the mod 2 cohomology of the classifying space of the spinor group spin(n). As a consequence of this computation, we show the non-collapsing of the Rothenberg-Steenrod spectral sequence for n > 16.

Borel cohomology and the relative Gorenstein condition for classifying spaces of compact Lie groups

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.

Cohomology of classifying spaces of loop groups and finite Chevalley groups associated with spin 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).

The equivariant simplicial de Rham complex and the classifying space of a semi-direct product group

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.

Equivariant local coefficients and the RO(G)-graded cohomology of classifying spaces

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)

We study the mod p cohomology of the classifying space of the projective unitary group PU(p). We first proof that old conjectures due to J.F. Adams, and Kono and Yagita about the structure of the mod p cohomology of classifying space of connected compact Lie groups held in the case of PU(p). Finally, we proof that the classifying space of the projective unitary group PU(p) is determined by its mod p cohomology as an unstable algebra over the Steenrod algebra for p>3, completing previous works for the cases p=2,3.

关于 $U(3)$ 交换性分类空间全空间的同调

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。

On the cohomology of classifying spaces of groups of homeomorphisms

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

We investigate the integral cohomology ring and the Chow ring of the classifying space of the complex projective linear group PGL_p, when p is an odd prime. In particular, we determine its additive structure completely, and we reduce the problem of determing its multiplicative structure to a problem in invariant theory.

The $\mathrm{RO}(G)$-Graded Cohomology of the Equivariant Classifying Space $B_G\mathrm{SU}(2)$

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…

子群族的分类空间的有界上同调

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。

关于 PUn 的分类空间上同调的 p-准素子群

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。

Assigning a classifying space to a fusion system up to F-isomorphism

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.

关于S^2上SO(n)-规范群的分类空间的上同调

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。

Integral cohomology of configuration spaces of the sphere

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.

等变实循环类映射与分类空间的Witt-sheaf上同调

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。

On the Cohomology of the Classifying Spaces of Projective Unitary Groups

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…

Classifying spaces for commutativity of low-dimensional Lie groups

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

The theory of secondary chomology operations leads to a conjecture concerning the algebra of higher cohomology operations in general. This conjecture is discussed here in detail and its connection with homotopy groups of spheres and the Adams spectral sequence is described.

Supersymmetric field theories and cohomology

This is the Ph.D. dissertation of the author. The project has been motivated by the conjecture that the Hopkins-Miller tmf spectrum can be described in terms of `spaces' of conformal field theories. In this dissertation, spaces of field theories are constructed as classifying spaces of categories whose objects are certain types of field theories. If such a category has a symmetric monoidal structure and its components form a group, by work of Segal, its classifying space is an infinite loop space and defines a cohomology theory. This has been carried out for two classes of field theories: (i) …

Classifying spaces of twisted loop groups

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

The class of loop spaces whose mod p cohomology is Noetherian is much larger than the class of p-compact groups (for which the mod p cohomology is required to be finite). It contains Eilenberg-Mac Lane spaces such as the infinite complex projective space and 3-connected covers of compact Lie groups. We study the cohomology of the classifying space BX of such an object and prove it is as small as expected, that is, comparable to that of BCP^\infty. We also show that BX differs basically from the classifying space of a p-compact group in a single homotopy group. This applies in particular to 4-c…

Cohomology with twisted coefficients of the classifying space of a fusion system

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 …

Eilenberg-Moore spectral sequence and Hodge cohomology of classifying stacks

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…

On the length of cohomology spheres

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.

纤维化分类空间的代数模型

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。

Weyl group invariants

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 $p=3$, $G=E_8$.

射影空间与4-球面乘积上自由对合的轨道空间的上同调代数

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。

在球面上的质量分割问题中的Bredon上同调方法

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。

偶 Clifford 群的 Nisnevich 分类空间的动机上同调

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。

Polynomial generators of msu*[1/2] related to classifying maps of certain formal group laws

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.

The rational cohomology Hopf algebra of a generic Kac-Moody group

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

Let p, ell be distinct primes and let q be a power of p. Let G be a connected compact Lie group. We show that there exists an integer b such that the mod ell cohomology of the classifying space of a finite Chevalley group G(F_q) is isomorphic to the mod ell cohomology of the classifying space of the loop group LG for q=p^{ab}, a>0.

The splitting of cohomology of p-groups with rank 2

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.

球面的Hopf等变映射空间的同调

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。

关于球面相对 Gelfand-Fuks 上同调的一个注记

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。

On the calculation of the cohomology of the third finite subset space of spheres

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.

相交上同调中的自然运算

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

We define secondary theories and characteristic classes for simplicial smooth manifolds generalizing Karoubi's multiplicative K-theory and multiplicative cohomology groups for smooth manifolds. As a special case we get versions of the groups of differential characters of Cheeger and Simons for simplicial smooth manifolds. Special examples include classifying spaces of Lie groups and Lie groupoids.

The Classifying Space of a Topological 2-Group

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…

Cohomology Classification of Spaces with Free S3 Actions

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

Let p be an odd prime. Let G be a p-local finite group over the extraspecial p-group p_+^{1+2}. In this paper we study the cohomology and the stable splitting of their p-complete classifying space BG.

The ER(2)-cohomology of X^nCP^\infty and BU(n)

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.

球面乘积连通和的同伦自等价的稳定上同调

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。

Chow-Witt rings of classifying spaces for symplectic and special linear groups

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…

Closed manifolds admitting no special generic maps whose codimensions are negative and their cohomology rings

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…

Fixed point Free Actions of Spheres and Equivariant maps

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…

Sullivan minimal models of classifying spaces for non-formal spaces of small rank

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.

Free actions of connected Lie groups of rank one on certain spaces

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.

三个球面乘积上的自由圆作用

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。

New families of manifolds with similar cohomology rings admitting special generic maps

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 cyclic group actions on some polyhedral products

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