Varietal isologism and covering groups

In this paper it will be shown that any two bfcal V-covering groups of a given group are Vbfcal V-isologic with respect to the variety Vbfcal V, which is a vast generalization of a result in B. Huppert (1967) and R. L. Griess JR (1973). We also give a criterion of existence of Vbfcal V-covering groups for a Vbfcal V-perfect group, and show that every automorphism of a given Vbfcal V-perfect group G can be extended to an automorphism of the Vbfcal V-covering G^* say, of G, this generalizes a result of J. L. Alperin and D. Gorenstein (1966), in the abelian variety.     

On the colored Jones polynomial, sutured Floer homology, and knot Floer homology

Let K⊂S³, and let denote the preimage of K inside its double branched cover, Σ(S³,K). We prove, for each integer n>1, the existence of a spectral sequence whose E² term is Khovanov's categorification of the reduced n-colored Jones polynomial of (mirror of K) and whose E^∞ term is the knot Floer homology of (when n odd) and of (S³,K#Kr) (when n even). A corollary of our result is that Khovanov's categorification of the reduced n-colored Jones polynomial detects the unknot whenever n>1.     

Covers of some varietal products

Received November 3, 1995; accepted in final form September 9, 1996.     

Transitivity of varietal hypercube networks

The varietal hypercube VQn is a variant of the hypercube Qn and has better properties than Qn with the same number of edges and vertices. This paper proves that VQn is vertex-transitive. This property shows that when VQn is used to model an interconnection network, it is high symmetrical and obviously superior to other variants of the hypercube such as the crossed cube.     

Characterization of varietal covering and stem groups

Let A _m be the variety of abelian groups of exponent m, where mis a square free positive integer. It is shown that every group G has an A _m -covering group, which generalizes the work of I. Schur (1904) and M.R. Jones (1973). Also, similar to the work of Yamazaki (1964) in the variety of abelian groups, if 1→ A → H → G → 1 is an A _m -stem extension, then we show that H is a homomorphic image of an A _m -covering group of G. Finally,in this variety some results of Hall's type will be proved.     

Double power laws,fractals and self-similarity

Power law (PL) distributions have been largely reported in the modeling of distinct real phenomena and have been associated with fractal structures and self-similar systems. In this paper, we analyze real data that follows a PL and a double PL behavior and verify the relation between the PL coefficient and the capacity dimension of known fractals. It is to be proved a method that translates PLs coefficients into capacity dimension of fractals of any real data.     

Ultrametric logarithm laws, II

We prove positive characteristic versions of the logarithm laws of Sullivan and Kleinbock–Margulis and obtain related results in metric diophantine approximation.     

变种超方体网络的容错直径与宽直径

研究变种超方体的网络容错直径和宽直径,证明了n维变种超立方体的n-1容错直径和n宽直径为[2n/3]+1或[2n/3]+2.     

Measure homology and singular homology are isometrically isomorphic

Measure homology is a variation of singular homology designed by Thurston in his discussion of simplicial volume. Zastrow and Hansen showed independently that singular homology (with real coefficients) and measure homology coincide algebraically on the category of CW-complexes. It is the aim of this paper to prove that this isomorphism is isometric with respect to the ℓ¹-seminorm on singular homology and the seminorm on measure homology induced by the total variation. This, in particular, implies that one can calculate the simplicial volume via measure homology – as already claimed by Thurston. For example, measure homology can be used to prove Gromov's proportionality principle of simplicial volume.     

Novikov homology,jump loci and Massey products

Let X be a finite CW complex, and ρ: π ₁(X) → GL(l, ?) a representation. Any cohomology class α ∈ H ¹(X, ?) gives rise to a deformation γ _t of ρ defined by γ _t (g) = ρ(g) exp(t〈α, g〉). We show that the cohomology of X with local coefficients γ _gen corresponding to the generic point of the curve γ is computable from a spectral sequence starting from H*(X, ρ). We compute the differentials of the spectral sequence in terms of the Massey products and show that the spectral sequence degenerates in case when X is a Kähler manifold and ρ is semi-simple. If α ∈ H ¹(X, ?) one associates to the triple (X, ρ, α) the twisted Novikov homology (a module over the Novikov ring). We show that the twisted Novikov Betti numbers equal the Betti numbers of X with coefficients in the local system γ _gen. We investigate the dependence of these numbers on α and prove that they are constant in the complement to a finite number of proper vector subspaces in H ¹(X, ?).     

Fundamental groups,homology equivalences and one-sided h-cobordisms

We give a necessary and sufficient condition for the fundamental group homomorphism of a map between CW-complexes(manifolds) to induce partial homology equivalences. As applications, we obtain characterizations of fundamental groups of homology spheres and Moore manifolds. Moreover, a classification of one-sided h-cobordism of manifolds up to diffeomorphisms is obtained, based on Quillen's plus construction with Whitehead torsions.     

On configuration spaces, their homology, and Lie algebras

Homological and geometric properties of configuration spaces of points in Euclidean space are described. Connections to non-abelian analogues of exterior algebras and mapping class groups are also given.