Craig A Jensen, Jon McCammond, John Meier
Let PSigma_n denote the group that can be thought of either as the group of motions of the trivial n-component link or the group of symmetric automorphisms of a free group of rank n. The integral cohomology ring of PSigma_n is determined, establishing a conjecture of Brownstein and Lee.
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Under mild conditions on the space X, we describe the additive structure of the integral cohomology of the space $X^p \times_{C_p}EC_p$ in terms of the cohomology of X. We give weaker results for other similar spaces, and deduce various corollaries concerning the cohomology of finite groups.
Integral cohomology of classical Artin groups: spherical types $A,C,D$ and affine types $\widetilde B,\widetilde C,\widetilde D$
本文利用代数Salvetti复形和Fuchs复形，描述了除仿射族$\widetilde A$外的无限经典族Artin群的整上同调代数加法结构。
It is not known whether or not the stable rational cohomology groups $\tilde H^*(Aut(F_\infty);\Q)$ always vanish. We show that either the rational cohomology does not vanish in certain dimensions, or the integral cohomology of a moduli space of pointed graphs does not stabilize in certain other dimensions. Similar results are stated for groups of outer automorphisms. This yields that $H^5(\hat Q_m; \mathbb{Z})$, $H^6(\hat Q_m; \mathbb{Z})$, and $H^5(Q_m; \mathbb{Z})$ never stabilize as $m \to \infty$, where the moduli spaces $\hat Q_m$ and $Q_m$ are the quotients of the spines $\hat X_m$ and …
We show that for topological groups and loop contractible coefficients the cohomology groups of continuous group cochains and of group cochains that are continuous on some identity neighbourhood are isomorphic. Moreover, we show a similar statement for compactly generated groups and Lie groups holds and apply our results to different concepts of group cohomology for finite-dimensional Lie groups.
We compute the integral cohomology rings of a family of 3-groups. As a corollary, we exhibit, for each n greater than or equal to 5, a pair of groups of order 3^n whose integral cohomology rings are isomorphic.
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.
We calculate the integral cohomology ring structure for various members of the Bianchi group family. The main tools we use are the Bockstein spectral sequence and a long exact sequence derived from Bass-Serre theory.
According to a result of Ehresmann, the torsions of integral homology of real Grassmannian are all of order $2$. In this note, We compute the $\mathbb{Z}_2$-dimensions of torsions in the integral homology and cohomology of real Grassmannian.
We determine the integral cohomology ring of the homogeneous space E_8/T^1E_7 by the Borel presentation and a method due to Toda. Then using the Gysin exact sequence associated with the circle bundle S^1 -> E_8/E_7 -> E_8/T^1E_7, we also determine the integral cohomology of E_8/E_7.
We determine the $T$-equivariant integral cohomology of $F_4/T$ combinatorially by the GKM theory, where T is a maximal torus of the exceptional Lie group $F_4$ and acts on $F_4/T$ by the left multiplication.
Integral bases for the second degree cohomology of 4-dimensional toric orbifolds
研究四维实维拟环面的上同调并获得其二次等价上同调的整数基，同时提供代数细胞基的另一种构造方法，并确定拟环面的Cartier除数群和Picard群。
We determine the integral cohomology rings of an infinite family of p-groups, for odd primes p, with cyclic derived subgroups. Our method involves embedding the groups in a compact Lie group of dimension one, and was suggested by P H Kropholler and J Huebschmann.
We give a complete description of the integral cohomology ring of the flag manifold E_8/T, where E_8 denotes the compact exceptional Lie group of rank 8 and T its maximal torus, by the method due to Borel and Toda. This completes the computation of the integral cohomology rings of the flag manifolds for all compact connected simple Lie groups.
We study the cohomology of spaces of string links and braids in $\mathbb{R}^n$ for $n\geq 3$ using configuration space integrals. For $n>3$, these integrals give a chain map from certain diagram complexes to the deRham algebra of differential forms on these spaces. For $n=3$, they produce all finite type invariants of string links and braids.
We prove the equivariant Leray-Hirsch theorem combinatorially for sufficiently good torus equivariant fiber bundles consisting of homogeneous spaces of Lie groups. We apply this theorem to determining the equivariant integral cohomology ring of the flag manifold of type $E_6$ and express it explicitly as a quotient ring of a polynomial ring.
Given a simply connected space $X$ with the cohomology $H^*(X;{\mathbb Z}_2)$ to be polynomial, we calculate the loop cohomology algebra $H^*(ΩX;{\mathbb Z}_2)$ by means of the action of the Steenrod cohomology operation $Sq_1$ on $H^*(X;{\mathbb Z}_2).$ As a consequence we obtain that $H^*(ΩX;{\mathbb Z}_2)$ is the exterior algebra if and only if $Sq_1$ is multiplicatively decomposable on $H^{\ast}(X;{\mathbb Z}_2).$ The last statement in fact contains a converse of a theorem of A. Borel.
Secondary cohomology operations and the loop space cohomology
本文受环路空间上同调启发，在单连通空间上构造次级上同调运算，使上同调成为Hopf代数，并用生成元和关系式计算环路空间上同调环，回答了Borel分解问题，应用于例外群$F_4$。
This note explores the interaction between cohomology operations in a generalized cohomology theory and a string topology loop coproduct dual to the Chas--Sullivan loop product. More precisely, we ask for a description for the failure of a given operation to commute with the loop coproduct, and will obtain a satisfactory answer in the case where the operation preserves both sums and products. Examples of such operations include the total Steenrod square in ordinary mod 2 cohomology and the Adams operations in K-theory.
On integral cohomology algebra of some oriented Grassmann manifolds
本文利用Leray-Serre谱序列完全确定了$\widetilde G_{8,3}$和$\widetilde G_{10,3}$的整系数上同调代数，并以奇数$n$的$\widetilde G_{n,2}$为例说明了该方法的适用性。
For each prime p, we exhibit pairs of p-groups all of whose integral cohomology groups are isomorphic. The method used involves very little calculation. The groups are exhibited as kernels of homomorphisms from a compact Lie group G to U(1), and the main result is that kernels of `similar' elements of Hom(G,U(1)) have isomorphic integral cohomology groups. The 2-groups constructed in this version have been corrected (there was a mistake in the presentations given in the published paper).
We construct E-infinity cell algebra models for the cochain algebras of the free and based loop spaces on a simply-connected topological space. Techniques from rational homotopy theory are exploited throughout.
Using methods applied by Atiyah in equivariant K-theory, Bredon obtained exact sequences for the relative cohomologies (with rational coefficients) of the equivariant skeletons of (sufficiently nice) T-spaces, T=(S^1)^n, with free equivariant cohomology over the cohomology of BT. Here we characterise those finite T-CW complexes with connected isotropy groups for which an analogous result holds with integral coefficients.
For each odd prime p, we exhibit p-groups G of p-rank two such that (suitably defined) Chern classes of unitary representations of G fail to generate the following rings: 1. The even degree integral cohomology of G; 2. The final page of the Atiyah-Hirzebruch spectral sequence for G; 3. The Brown-Peterson generalized cohomology of G. It follows that these groups afford counterexamples to conjectures of C. B. Thomas, M. F. Atiyah and P. Landweber.
The Hilbert scheme X^{[a]} of points on a complex manifold X is a compactification of the configuration space of a-element subsets of X. The integral cohomology of X^{[a]} is more subtle than the rational cohomology. In this paper, we compute the mod 2 cohomology of X^{[2]} for any complex manifold X, and the integral cohomology of X^{[2]} when X has torsion-free cohomology. The results of this paper are used in Voisin's work on the universal CH_0 group of cubic hypersurfaces, because the crucial point there is to study the 2-torsion in the Chow group.
We construct a complex analytic version of an equivariant cohomology theory which appeared in a recent paper of Rezk, and which is roughly modeled on the Borel-equivariant cohomology of the double free loop space. The construction is defined on finite, torus-equivariant CW complexes and takes values in coherent holomorphic sheaves over the moduli stack of complex elliptic curves. Our methods involve an inverse limit construction over all finite dimensional subcomplexes of the double free loop space, following an analogous construction of Kitchloo for single free loop spaces. We show that, for …
The 11-loop graph cohomology
本文计算了圈数为11及更高圈数部分度数的Kontsevich图同调，并据此对F. Brown猜想的一个强版本给出反例。
We compute the integral homology and cohomology groups of configuration spaces of two distinct points on a given real projective space. The explicit answer is related to the (known multiplicative structure in the) integral cohomology---with simple and twisted coefficients---of the dihedral group of order 8 (in the case of unordered configurations) and the elementary abelian 2-group of rank 2 (in the case of ordered configurations). As an application, we complete the computation of the symmetric topological complexity of real projective spaces of dimension 2^i + d for d=0,1,2.
We present a method to compute integral cohomology of posets. This toolbox is applicable as soon as the sub-posets under each object possess certain structure. This is the case for simplicial complexes and simplex-like posets. The method is based on homological algebra arguments in the category of functors and on a spectral sequence built upon the poset. We show its relation to discrete Morse theory. As application we give an alternative proof of Webb's conjecture for saturated fusion systems and we compute the cohomology of Coxeter complexes for finite and infinite Coxeter groups.
Let G be an exceptional Lie group with a maximal torus T. Based on Schubert calculus on the flag manifold G/T we have described the integral cohomology ring H*(G) by explicitely constructed generators in [DZ2], and determined the structure of H*(G;F_{p}) as a Hopf algebra over the Steenrod algebra in[DZ3]. In this sequel to [DZ2], [DZ3] we obtain the near--Hopf ring structure on H*(G).
Cohomology rings of extended powers and free infinite loop spaces
本文计算了扩展幂及其群完备化的模p上同调，通过识别带有除幂的Hopf环结构简化杯积计算，并基于对称群上同调结果，进一步理解无穷扩展幂与自由无穷环路空间的上同调环。
Let $ϕ:\Z/p\to GL_{n}(\Z)$ denote an integral representation of the cyclic group of prime order $p$. This induces a $\Z/p$-action on the torus $X=\R^{n}/\Z^{n}$. The goal of this paper is to explicitly compute the cohomology groups $H^{*}(X/\Z/p;\Z)$ for any such representation. As a consequence we obtain an explicit calculation of the integral cohomology of the classifying space associated to the family of finite subgroups for any crystallographic group $Γ=\Z^n\rtimes\Z/p$ with prime holonomy.
The Borel Cohomology of Free Iterated Loop Spaces
本文研究了自由迭代循环空间的Borel上同调，通过构造特定映射来计算模2上同调，并在n=1时恢复了已知结果。
Let X be a simply connected space and k a commutative ring. Goodwillie, Burghelea and Fiedorowiscz proved that the Hochschild cohomology of the singular chains on the pointed loop space HH^{*}S_*(ΩX) is isomorphic to the free loop space cohomology H^{*}(X^{S^{1}}). We proved that this isomorphism is compatible with both the cup product on HH^{*}S_*(ΩX) and on H^{*}(X^{S^{1}}). In particular, we explicit the algebra H^{*}(X^{S^{1}}) when X is a suspended space, a complex projective space or a finite CW-complex of dimension p such that \frac {1}{(p-1)!}\in k.
`Loop-fusion cohomology' is defined on the continuous loop space of a manifold in terms of \vCech cochains satisfying two multiplicative conditions with respect to the fusion and figure-of-eight products on loops. The main result is that these cohomology groups, with coefficients in an abelian group, are isomorphic to those of the manifold and the transgression homomorphism factors through the isomorphism.
We apply the Transfer Algorithm introduced in arXiv:1106.5090 to transfer an A_\infty-algebra structure that cannot be computed using the classical Basic Perturbation Lemma. We construct a space X whose (base pointed) loop cohomology H = H^*(ΩX; Z_2) comes equipped with a nontrivial operation ω: H x H --> H x H.
Integral cohomology rings of weighted Grassmann orbifolds and rigidity properties
本文研究了带权格拉斯曼 orbifold 的整数上同调环及其刚性性质，通过普吕克权重向量进行分类并计算了其上同调环。
In this paper, we find weight decomposition and rank of a weight in the integral (co)homology of the positive system of a semi-simple Lie algebra over $\Bbb C$ and prove that the (co)homology of the weight subcomplex over a field of characteristic p is 0 if the rank of the weight is not divisible by p. This generalizes Kostant theorem to the integral cohomology of the positive system.
In this article, we show the existence of conjugations on many simply-connected spin 6-manifolds with free integral cohomology. In a certain class the only condition on X^6 to admit a conjugation with fixed point set M^3 is the obvious one: the existence of a degree-halving ring isomorphism between the Z_2-cohomologies of X and M.
Let G be a compact connected Lie group with a maximal torus T\subsetG. In the context of Schubert calculus we obtain a canonical presentation for the integral cohomology ring H^{\ast}(G/T) of the complete flag manifold G/T. The result have been applied in [15] to construct the integral cohomology ring H^{\ast}(G) in terms of Schubert classes on G/T, and in [16] to determine the structure of the modp cohomology H^{\ast}(G;F_{p}) as a Hopf algebra over the Steenrod algebra.
We show that if X is a smooth complex projective surface with torsion-free cohomology, then the Hilbert scheme X^[n] has torsion-free cohomology for every natural number n. This extends earlier work by Markman on the case of Poisson surfaces. The proof uses Gholampour-Thomas's reduced obstruction theory for nested Hilbert schemes of surfaces.
The integral cohomology rings of Peterson varieties in type A
本文研究 A_{n-1} 型 Peterson 簇的整上同调环，分别利用 permutohedral 簇的 S_n 不变子环以及生成元与关系给出其环结构的两种描述。
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).
A complete flag manifold is the quotient of a Lie group by its maximal torus. The rank of a flag manifold is the dimension of the maximal torus of the Lie group. The rank 2 complete flag manifolds are $SU(3)/T^2$, $Sp(2)/T^2$, $Spin(4)/T^2$, $Spin(5)/T^2$ and $G_2/T^2$. In this paper we calculate the cohomology of the free loop space of rank 2 complete flag manifolds.
In this paper we study the integral cohomology of pure mapping class groups of surfaces, and other related groups and spaces, as FI-modules. We use recent results from Church, Miller, Nagpal and Reinhold to obtain explicit linear bounds for their presentation degree and to give an inductive description of these FI-modules. Furthermore, we establish new results on representation stability, in the sense of Church and Farb, for the rational cohomology of pure mapping class groups of non-orientable surfaces.
The integral cohomology rings of four-dimensional toric orbifolds
本文研究四维托里克 orbifold 的整数上同调环，通过构造显式基底并表达基底元素的杯积，确定了上同调环的结构。
The CW structure of certain spaces, such as effective orbifolds, can be too complicated for computational purposes. In this paper we use the concept of $\mathbf{q}$-CW complex structure on an orbifold, to detect torsion in its integral cohomology. The main result can be applied to well known classes of orbifolds or algebraic varieties having orbifold singularities, such as toric orbifolds, simplicial toric varieties, torus orbifolds and weighted Grassmannians.
4-torsion classes in the integral cohomology of oriented Grassmannians
该研究探讨定向格拉斯曼流形整系数上同调的4-挠类，基于Steenrod平方给出存在判据，推进特征秩猜想的证明，提供无穷多4-挠实例，并提出齐性空间挠指数与亏格的猜想关系。
This paper proves an integral version of the Riemann-Roch theorem for surface bundles, comparing the standard cohomology classes with the cohomology classes coming from the symplectic group.
We study the cohomology of the free loop space of $SU(n+1)/T^n$, the simplest example of a complete flag manifolds and an important homogeneous space. Through this enhanced analysis we reveal rich new combinatorial structures arising in the cohomology algebra of the free loop spaces. We build new theory to allow for the computation of $H^*(Λ(SU(n+1)/T^{n});\mathbb{Z})$, a significantly more complicated structure than other known examples. In addition to our theoretical results, we explicitly implement a novel integral Gröbner basis procedure for computation. This procedure is applicable to any…
We compute the integral cohomology ring of configuration spaces of two points on a given real projective space. Apart from an integral class, the resulting ring is a quotient of the known integral cohomology of the dihedral group of order 8 (in the case of unordered configurations, thus has only 2- and 4-torsion) or of the elementary abelian 2-group of rank 2 (in the case of ordered configurations, thus has only 2-torsion). As an application, we complete the computation of the symmetric topological complexity of real projective spaces of dimensions of the form 2^i+j for non-negative i and j wi…
Let $X_Σ$ be a smooth, not necessarily compact toric variety. We show that a certain complex, defined in terms of the fan $Σ$, computes the integral cohomology of $X_Σ$, including the module structure over the homology of the torus. In some cases we can also give the product. As a corollary we obtain that the cycle map from Chow groups to integral Borel-Moore homology is split injective for smooth toric varieties. Another result is that the differential algebra of singular cochains on the Borel construction of $X_Σ$ is formal.
On the integral cohomology of real toric manifolds
本文研究实 toric 流形的整数上同调，通过组合数据推导出上同调群公式，并提出环结构的组合公式，进一步给出判断 spin^c 流形的准则。
Equivariant elliptic cohomology and the 2-loop groupoid
本文遵循 Rezk 的提纲，为任意紧 Lie 群构造复解析等变椭圆上同调，证明其基本性质，并以二维环面主丛群胚的轨道范畴参数化，推广 Grojnowski 的构造。
In this paper we identify conditions under which the cohomology $H^*(ΩMξ;\k)$ for the loop space $ΩMξ$ of the Thom space $Mξ$ of a spherical fibration $ξ\downarrow B$ can be a polynomial ring. We use the Eilenberg-Moore spectral sequence which has a particularly simple form when the Euler class $e(ξ)\in H^n(B;\k)$ vanishes, or equivalently when an orientation class for the Thom space has trivial square. As a consequence of our homological calculations we are able to show that the suspension spectrum $Σ^\inftyΩMξ$ has a local splitting replacing the James splitting of $ΣΩMξ$ when $Mξ$ is a susp…
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.
The goal of this article is to make explicit a structured complex whose homology computes the cohomology of the p-profinite completion of the n-fold loop space of a sphere of dimension d=n-m<n. This complex is defined purely algebraically, in terms of characteristic structures of E_n-operads. Our construction involves: the free complete algebra in one variable associated to any E_n-operad; and an element in this free complete algebra, which is associated to a morphism from the operad of L-infinity algebras to an operadic suspension of our E_n-operad. We deduce our main theorem from: a connecti…
We study natural subalgebras Ch_E(G) of group cohomology defined in terms of infinite loop spaces E and give representation theoretic descriptions of those based on QS^0 and the Johnson-Wilson theories E(n). We describe the subalgebras arising from the Brown-Peterson spectra BP and as a result give a simple reproof of Yagita's theorem that the image of BP^*(BG) in H^*(BG;F_p) is F-isomorphic to the whole cohomology ring; the same result is shown to hold with BP replaced by any complex oriented theory E with a map of ring spectra from E to HF_p which is non-trivial in homotopy. We also extend t…
Let $G$ be a compact Lie group with a maximal torus $T$. Based on a presentation of the integral cohomology ring $H^{\ast}(G/T)$ of the flag manifold $G/T$ in \cite{DZ1}we have presented in \cite{DZ2}an explicit and unified construction of the integral cohomology rings $H^{\ast}(G)$ for the $1$--connected Lie groups $G$. In this paper we extend this construction to all compact Lie groups.
We give a lower bound for the exponent of certain elements in the integral cohomology of the total spaces of principal BC-bundles for C a finite cyclic group. As applications we give a proof of the theorem of A. Adem and H.-W. Henn that a p-group is elementary abelian if and only if its integral cohomology has exponent p, and we exhibit some infinite groups of finite virtual cohomological dimension whose Tate-Farrell cohomology contains torsion of order greater than the l.c.m. of the orders of their finite subgroups. We also give an upper bound for the exponent of all but finitely many of the …
We study the mod-2 cohomology spectral sequence arising from delooping the Bousfield-Kan cosimplicial space giving the 2-nilpotent completion of a connective spectrum $X$. Under good conditions its $E_{2}$-term is computable as certain non-abelian derived functors evaluated at $H^*(X)$ as a module over the Steenrod algebra, and it converges to the cohomology of $Ω^\infty X$. We provide general methods for computing the $E_{2}$-term, including the construction of a multiplicative spectral sequence of Serre type for cofibration sequences of simplicial commutative algebras. Some simple examples a…
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