Related papers: Generators for Group Homology and a Vanishing Conj…

We determine the minimal number of generators of the homological Goldman Lie algebra of a surface consisting of elements of the first homology group of the surface.

For any (Hausdorff) compact group $G$ with the normalized Haar measure ${\mathbf m}_G$, denote by ${\rm cp}(G)$ the probability ${\mathbf m}_{G\times G}(\{(x,y)\in G\times G \;|\; xy=yx\})$ of commuting a randomly chosen pair of elements of…

Let $G$ be a finite group generated by $k$ elements. The well-known product replacement algorithm provides an effective method for sampling generating sets of $G$. We study a refinement of this algorithm that is designed to output…

The codomain category of a generalized homology theory is the category of modules over a ring. For an abelian category A, an A-valued (generalized) homology theory is defined by formally replacing the category of modules with the category…

The notion of Hopf center and Hopf cocenter of a Hopf algebra is investigated by the extension theory of Hopf algebras. We prove that each of them yields an exact sequence of Hopf algebras. Moreover the exact sequences are shown to satisfy…

We give an elementary proof that there are two topological generators for the full group of every aperiodic hyperfinite probability measure preserving Borel equivalence relation. Our proof explicitly constructs topological generators for…

A set of ring generators for the Hecke algebra of the Gel'fand pair $(S_{2n},B_n)$, where $B_n$ is the hyperoctahedral subgroup of the symmetric group $S_{2n}$ is presented. Various corollaries are given. A conjecture of Sho Matsumoto is…

Let $G$ be a finite group. In a famous article, Quillen describes an $\mathrm{F}$-isomorphism between commutative $\mathbb{N}$-graded $\mathbb{F}_{2}$-algebras $$\mathrm{q}_{G}:\mathrm{H}^{*}(G;\mathbb{F}_{2})\to\mathrm{L}(G)\ ,$$ with…

We give a new proof of Quillen's conjecture for solvable groups via a geometric and explicit method. For p-solvable groups, we provide both a new proof using the Classification of Finite Simple Groups and an asymptotic version without…

If $p$ is an odd prime, then we prove that $\e(H_2(G,\mathbb{Z})) \mid p\ \e(G)$ for $p$ groups of class 7. We prove the same for $p$ groups of class at most $p+1$ with $\e(Z(G))=p$. We also prove Schurs conjecture if $\e(G/Z(G))$ is $2,3$…

A generating function of the number of homomorphisms from the fundamental group of a compact oriented or non-orientable surface without boundary into a finite group is obtained in terms of an integral over a real group algebra. We calculate…

We generalize the Plesken-Fabia\'nska $\mathrm{L}_2$-quotient algorithm for finitely presented groups on two or three generators to allow an arbitrary number of generators. The main difficulty lies in a constructive description of the…

In this paper we prove that every $2$-generator finite $p$-group $G$ has a non-inner automorphism of order $p$ leaving $G^p\gamma_4(G)$ elementwise fixed ($p\ge 5$). Moreover, we prove a $2$-generator finite $3$-group satisfying…

We calculate the second rational homology group of the Torelli group for $g \geq 6$.

In a previous paper, the author (together with Matthew Emerton) proved that the completed cohomology groups of SL_N(Z) are stable in fixed degree as N goes to infinity (Z may be replaced by the ring O_F of integers of any number field). In…

We say that a space X admits a homology exponent if there exists an exponent for the torsion subgroup of the integral homology. Our main result states if an H-space of finite type admits a homology exponent, then either it is, up to…

The goal of this paper is to establish fundamental properties of the Hochschild, topological Hochschild, and topological cyclic homologies of commutative, Noetherian rings, which are assumed only to be F-finite in the majority of our…

We give a systematic description of the cyclic cohomology theory of Hopf algebroids in terms of its associated category of modules. Then we introduce a dual cyclic homology theory by applying cyclic duality to the underlying cocyclic…

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…

The finite dual $H^{\circ}$ of an affine commutative-by-finite Hopf algebra $H$ is studied. Such a Hopf algebra $H$ is an extension of an affine commutative Hopf algebra $A$ by a finite dimensional Hopf algebra $F$. The main theorem gives…