Profinite groups, profinite completions and a conjecture of Moore

Let R be any ring (with 1), Γ a group and RΓ the corresponding group ring. Let H be a subgroup of Γ of finite index. Let M be an RΓ-module, whose restriction to RH is projective.Moore's conjecture (J. Pure Appl. Algebra 7(1976)287): Assume for every nontrivial element x in Γ, at least one of the following two conditions holds:

(M1)

〈x〉∩H≠{e} (in particular this holds if Γ is torsion free)

(M2)

ord(x) is finite and invertible in R.

Then M is projective as an RΓ-module.More generally, the conjecture has been formulated for crossed products R*Γ and even for strongly graded rings R(Γ). We prove the conjecture for new families of groups, in particular for groups whose profinite completion is torsion free.The conjecture can be formulated for profinite modules M over complete groups rings [[RΓ]] where R is a profinite ring and Γ a profinite group. We prove the conjecture for arbitrary profinite groups. This implies Serre's theorem on cohomological dimension of profinite groups.     

Hyperbolic spaces at large primes and a conjecture of Moore

A simply connected finite complex X is called elliptic if its rational homotopy Lie algebra is of finite dimension and hyperbolic otherwise. According to a conjecture of Moore, there exists an exponent for the p-torsion part of if and only if X is elliptic. In this note, it is shown that, provided the prime p is sufficiently large, a hyperbolic space with p-torsion free loop space homology has no exponent in the p-torsion of the homotopy groups. For a class of formal spaces, this result is obtained for every odd prime.     

Propagating terraces in a proof of the Gibbons conjecture and related results

The Gibbons conjecture stating the one-dimensional symmetry of certain solutions of semilinear elliptic equations has been proved by several authors. We show how attractivity properties of minimal propagating terraces of one-dimensional parabolic problems can be used in a proof of a version of this result and related statements.     

New results on the conjecture of Rhodes and on the topological conjecture

The Conjecture of Rhodes, originally called the “type II conjecture” by Rhodes, gives an algorithm to compute the kernel of a finite semigroup. This conjecture has numerous important consequences and is one of the most attractive problems on finite semigroups. It was known that the conjecture of Rhodes is a consequence of another conjecture on the finite group topology for the free monoid. In this paper, we show that the topological conjecture and the conjecture of Rhodes are both equivalent to a third conjecture and we prove this third conjecture in a number of significant particular cases.     

Nilpotency of Bocksteins, Kropholler's hierarchy and a conjecture of Moore

We show that the class of pairs (Γ,H) of a group and a finite index subgroup which verify a conjecture of Moore about projectivity of modules over ZΓ satisfy certain closure properties. We use this, together with a result of Benson and Goodearl, in order to prove that Moore's conjecture is valid for groups which belongs to Kropholler's hierarchy LHF. For finite groups, Moore's conjecture is a consequence of a theorem of Serre, about the vanishing of a certain product in the cohomology ring (the Bockstein elements). Using our result, we construct examples of pairs (Γ,H) which satisfy the conjecture without satisfying the analog of Serre's theorem.     

A counterexample to a conjecture related to the Jacobian problem

Translated fromMatematicheskie Zametki, Vol. 58, No. 3, pp. 452–455, September, 1995.     

Some remarks related to De Giorgi's conjecture

For several classes of functions including the special case , we obtain boundedness and symmetry results for solutions of the problem defined on . Our results complement a number of recent results related to a conjecture of De Giorgi.

     

On Bohman's conjecture related to a sum packing problem of Erdos

Motivated by a sum packing problem of Erdos, Bohman discussed an extremal geometric problem which seems to have an independent interest. Let be a hyperplane in such that . The problem is to determine

Bohman (1996) conjectured that

We show that for some constants we have --disproving the conjecture. We also consider a more general question of the estimation of , when , k>1$">.

     

On certain co-H spaces related to Moore spaces

We show that certain co- spaces, constructed by Anick and Gray, carry a homotopy co-associative and co-commutative co- structure.

     

A problem related to Foulkes’s conjecture

In this paper we study a class of symmetric matricesT indexed by positive integers m≥ n≥2 and defined as follows: for any positive integersp andq let ?_p,q be the set of partitions ofU = {1,2,3, ...,pq} into p blocks each of sizeq. Letm ≥n ≥ 2 be positive integers. By atransversal of α = A₁/A₂/.../A_n ∈ ?_n,m we mean a partitionß = B₁/B₂/.../B_m ∈ ? _m,n such that ‖A _i ∩B _j = 1 for every i= 1,2, ...,n and everyj = 1,2, ...,m. LetM be the zero-one matrix with rows indexed by the elements of ?_n,m and columns indexed by the elements of ?_m,n such that M_αß = 1 iffß is a transversal of α. We are interested in finding the eigenvalues and eigenspaces of the symmetric matrixT = MM^t. The nonsingularity ofT implies Foulkes’s Conjecture (for these values of m andn). In the casen = 2 we completely determine the eigenvalues and eigenspaces of T and in so doing demonstrate the non-singularity ofT. Forn = 3 we develop a fast algorithm for computing the eigenvalues ofT, and give numerical results in the cases m = 3,4, 5, 6.     

Vizing's conjecture: a survey and recent results

Vizing's conjecture from 1968 asserts that the domination number of the Cartesian product of two graphs is at least as large as the product of their domination numbers. In this paper we survey the approaches to this central conjecture from domination theory and give some new results along the way. For instance, several new properties of a minimal counterexample to the conjecture are obtained and a lower bound for the domination number is proved for products of claw‐free graphs with arbitrary graphs. Open problems, questions and related conjectures are discussed throughout the paper. © 2011 Wiley Periodicals, Inc. J Graph Theory 69: 46–76, 2012     

关于Reed列表染色猜想的一些结果

构造了一个图G,给G的每个顶点v一个颜色列表,使得每个列表Lv的大小至少为每个顶点v的邻域NG(v)与每个Vc交集的最大数目,但是这个图不存在一个正常的列表染色,从而推翻了R eed的一个猜想.     

Counting points on varieties over finite fields related to a conjecture of Kontsevich

We describe a characteristic-free algorithm for reducing an algebraic variety defined by the vanishing of a set of integer polynomials. In very special cases, the algorithm can be used to decide whether the number of points on a variety, as the ground field varies over finite fields, is a polynomial function of the size of the field. The algorithm is then used to investigate a conjecture of Kontsevich regarding the number of points on a variety associated with the set of spanning trees of any graph. We also prove several theorems describing properties of a (hypothetical) minimal counterexample to the conjecture, and produce counterexamples to some related conjectures.Partially supported by NSF Grant DMS-9700787 and RIMS, Kyoto University.     

A counterexample to the Alon-Saks-Seymour conjecture and related problems

Consider a graph obtained by taking an edge disjoint union of k complete bipartite graphs. Alon, Saks, and Seymour conjectured that such graphs have chromatic number at most k+1. This well known conjecture remained open for almost twenty years. In this paper, we construct a counterexample to this conjecture and discuss several related problems in combinatorial geometry and communication complexity.     

Combinatorial designs related to the strong perfect graph conjecture

When α, ω are positive integers, we set n = αω + 1 and look for zero-one matrices X, Y of size n × n such that

$\text{XY= YX = J ? I}$

,

$\text{JX = XJ = αJ}$

, JY = YJ = ωJ. Simple solutions of these matrix equations are easy to find; we describe ways of constructing rather messy ones. Our investigations are motivated by an intimate relationship between the pairs X, Y and minimal imperfect graphs.     

A periodic isoperimetric problem related to the unique games conjecture

We prove the endpoint case of a conjecture of Khot and Moshkovitz related to the unique games conjecture, less a small error. Let n ≥ 2. Suppose a subset Ω of n‐dimensional Euclidean space satisfies ?Ω = Ω^c and Ω + v = Ω^c (up to measure zero sets) for every standard basis vector . For any and for any q ≥ 1, let and let . For any x ∈ ?Ω, let N(x) denote the exterior normal vector at x such that ‖N(x)‖₂ = 1. Let . Our main result shows that B has the smallest Gaussian surface area among all such subsets Ω, less a small error: In particular, Standard arguments extend these results to a corresponding weak inequality for noise stability. Removing the factor 6 × 10^?9 would prove the endpoint case of the Khot‐Moshkovitz conjecture. Lastly, we prove a Euclidean analogue of the Khot and Moshkovitz conjecture. The full conjecture of Khot and Moshkovitz provides strong evidence for the truth of the unique games conjecture, a central conjecture in theoretical computer science that is closely related to the P versus NP problem. So, our results also provide evidence for the truth of the unique games conjecture. Nevertheless, this paper does not prove any case of the unique games conjecture.