Alejandro Adem, Jonathan Pakianathan
We use topological methods to compute the mod p cohomology of certain p-groups. More precisely we look at central Frattini extensions of elementary abelian by elementary abelian groups such that their defining k-invariants span the entire image of the Bockstein. We show that if p is sufficiently large, then the mod p cohomology of the extension can be explicitly computed as an algebra.
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Henn, Lannes, and Schwartz have introduced two invariants, d_0(G) and d_1(G), of the mod p cohomology of a finite group G such that H^*(G) is detected and determined by H^d(C_G(V)) for d no bigger than d_0(G) and d_1(G), with V < G p-elementary abelian. We study how to calculate these invariants. We define a number e(G) that measures the image of the restriction of H^*(G) to its maximal central p-elementary abelian subgroup. Using Benson--Carlson duality, we show that when $G$ has a p-central Sylow subgroup P, d_0(G) = d_0(P) = e(P), and a similar exact formula holds for d_1(G). In general, we…
Central Extensions and Cohomology
本文针对拓扑为CW复形的群G，将其由离散阿贝尔群A给出的中心扩张关联到分类空间BG的上同调类，证明该关联的单射性等性质，给出同态可提升为中心扩张的充要条件。
For stable degree zero operations, and also for additive unstable operations of bidegree (0,0), it is known that the centre of the ring of operations for complex cobordism is isomorphic to the corresponding ring of connective complex K-theory operations. Similarly, the centre of the ring of BP operations is the corresponding ring for the Adams summand of p-local connective complex K-theory. Here we show that, in the additive unstable context, this result holds with BP replaced by BP<n> for any n. Thus, for all chromatic heights, the only central operations are those coming from K-theory.
It was shown by Gersten that a central extension of a finitely generated group is quasi-isometrically trivial provided that its Euler class is bounded. We say that a finitely generated group $G$ satisfies Property QITB (quasi-isometrically trivial implies bounded) if the Euler class of any quasi-isometrically trivial central extension of $G$ is bounded. We exhibit a finitely generated group $G$ which does not satisfy Property QITB. This answers a question by Neumann and Reeves, and provides partial answers to related questions by Wienhard and Blank. We also prove that Property QITB holds for a…
We study H^*(P), the mod p cohomology of a finite p--group P, viewed as an Out(P)--module. In particular, we study the conjecture, first considered by Martino and Priddy, that, if e_S in Z/p[Out(P)] is a primitive idempotent associated to an irreducible Z/p[Out(P)]--module S, then the Krull dimension of e_SH^*(P) equals the rank of P. The rank is an upper bound by Quillen's work, and the conjecture can be viewed as the statement that every irreducible Z/p[Out(P)]--module occurs as a composition factor in H^*(P) with similar frequency. In summary, our results are as follows. A strong form of th…
For cohomology theories closely related to Morava E-theory, we provide an algebro-geometric interpretation of the cohomology of groups that arise as centralizers of tuples of commuting elements inside of symmetric groups. The interpretation is given in terms of the connected components of the scheme that classifies subgroup schemes of a particular p-divisible group.
We study the cohomology H*(A) = Ext_A(k,k) of a locally finite, connected, cocommutative Hopf algebra A over k = F_p. Specifically, we are interested in those algebras A for which H*(A) is generated as an algebra by H^1(A) and H^2(A). We shall call such algebras semi-Koszul. Given a central extension of Hopf algebras F --> A --> B with F monogenic and B semi-Koszul, we use the Cartan-Eilenberg spectral sequence and algebraic Steenrod operations to determine conditions for A to be semi-Koszul. Special attention is given to the case in which A is the restricted universal enveloping algebra of th…
In this paper we first show that many braid groups of low genus surfaces have their centers as direct factors. We then give a description of centralizers and normalizers of prime order elements in pure mapping class groups of surfaces with spherical quotients using automorphism groups of fundamental groups of the quotient surfaces. As an application, we use these to show that the $p$-primary part of the Farrell cohomology groups of certain mapping class groups are elementary abelian groups. At the end we compute the $p$-primary part of the Farrell cohomology of a few pure mapping class groups.
Mickelsson defined a group 2-cocycle on the group of smooth maps from the closed unit 2-disk to a Lie group G and constructed a smooth central extension on a loop group of G, called the affine Kac-Moody central extension. We reformulate this central extension theory in the terms of the seven-term exact sequence for cohomology groups for discrete groups.
For prime power $q=p^d$ and a field $F$ containing a root of unity of order $q$ we show that the Galois cohomology ring $H^*(G_F,\dbZ/q)$ is determined by a quotient $G_F^{[3]}$ of the absolute Galois group $G_F$ related to its descending $q$-central sequence. Conversely, we show that $G_F^{[3]}$ is determined by the lower cohomology of $G_F$. This is used to give new examples of pro-$p$ groups which do not occur as absolute Galois groups of fields.
We compute the continuous cohomology of the Morava stabilizer group with coefficients in Morava $E$-theory, $H^*(\mathbb{G}_2, E_t)$, at $p=2$, for $0\leq t < 12$, using the Algebraic Duality Spectral Sequence. Furthermore, in that same range, we compute the $d_3$-differentials in the homotopy fixed point spectral sequence for the $K(2)$-local sphere spectrum. These cohomology groups and differentials play a central role in $K(2)$-local stable homotopy theory.
Constructing the Virasoro groups using differential cohomology
本文利用第一 Pontryagin 类的非对角微分提升，对 Virasoro 群的中心扩张给出新颖的几何构造，并肯定回答了 Freed-Hopkins 的问题。
We explore two constructions in homotopy category with algebraic precursors in the theory of noncommutative rings and homological algebra, namely the Hochschild cohomology of ring spectra and Morita theory. The present paper provides an extension of the algebraic theory to include the case when $M$ is not necessarily a progenerator. Our approach is complementary to recent work of Dwyer and Greenlees and of Schwede and Shipley. A central notion of noncommutative ring theory related to Morita equivalence is that of central separable or Azumaya algebras. For such an Azumaya algebra A, its Hochsch…
A central extension of the form $E: 0 \to V \to G \to W \to 0$, where $V$ and $W$ are elementary abelian 2-groups, is called Bockstein closed if the components $q_i \in H^*(W, \FF_2)$ of the extension class of $E$ generate an ideal which is closed under the Bockstein operator. In this paper, we study the cohomology ring of $G$ when $E$ is a Bockstein closed 2-power exact extension. The mod-2 cohomology ring of $G$ has a simple form and it is easy to calculate. The main result of the paper is the calculation of the Bocksteins of the generators of the mod-2 cohomology ring using an Eilenberg-Moo…
Mod 2 magnitude cohomology ring of real hyperplane arrangements
本文针对有限中心实超平面排列的顶图，改进Koizumi的模长同调结果，利用典范上同调基确定其交叉分次模长上同调环，明确了直和项的秩性质。
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…
Let X and Y be finite-type CW-complexes (X connected, Y simply connected), such that the rational cohomology ring of Y is a k-rescaling of the rational cohomology ring of X. Assume H^*(X,Q) is a Koszul algebra. Then, the homotopy Lie algebra pi_*(Omega Y) tensor Q equals, up to k-rescaling, the graded rational Lie algebra associated to the lower central series of pi_1(X). If Y is a formal space, this equality is actually equivalent to the Koszulness of H^*(X,Q). If X is formal (and only then), the equality lifts to a filtered isomorphism between the Malcev completion of pi_1(X) and the complet…
We construct a global geometric model for complex analytic equivariant elliptic cohomology for all compact Lie groups. Cocycles are specified by functions on the space of fields of the two-dimensional sigma model with background gauge fields and $\mathcal{N} = (0, 1)$ supersymmetry. We also consider a theory of free fermions valued in a representation whose partition function is a section of a determinant line bundle. We identify this section with a cocycle representative of the (twisted) equivariant elliptic Euler class of the representation. Finally, we show that the moduli stack of $U(1)$-g…
Pattern-equivariant (PE) cohomology is a well-established tool with which to interpret the Čech cohomology groups of a tiling space in a highly geometric way. In this paper we consider homology groups of PE infinite chains. We establish Poincaré duality between the PE cohomology and PE homology. The Penrose kite and dart tilings are taken as our central running example, we show how through this formalism one may give highly approachable geometric descriptions of the generators of the Čech cohomology of their tiling space. These invariants are also considered in the context of rotational symmet…
Homology decomposition techniques are a powerful tool used in the analysis of the homotopy theory of (classifying) spaces. The associated Bousfield-Kan spectral sequences involve higher derived limits of the inverse limit functor. We study the impact of depth conditions on the vanishing of these higher limits and apply our theory in several cases. We will show that the depth of Stanley-Reisner algebras can be characterized in combinatorial terms of the underlying simplicial complexes, the depth of group cohomology in terms of depth of group cohomology of centralizers of elementary abelian subg…
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