Topology of random clique complexes

In a seminal paper, Erd?s and Rényi identified a sharp threshold for connectivity of the random graph G(n,p). In particular, they showed that if p?logn/n then G(n,p) is almost always connected, and if p?logn/n then G(n,p) is almost always disconnected, as n→∞.The clique complexX(H) of a graph H is the simplicial complex with all complete subgraphs of H as its faces. In contrast to the zeroth homology group of X(H), which measures the number of connected components of H, the higher dimensional homology groups of X(H) do not correspond to monotone graph properties. There are nevertheless higher dimensional analogues of the Erd?s-Rényi Theorem.We study here the higher homology groups of X(G(n,p)). For k>0 we show the following. If p=n^α, with α<−1/k or α>−1/(2k+1), then the kth homology group of X(G(n,p)) is almost always vanishing, and if −1/k<α<−1/(k+1), then it is almost always nonvanishing.We also give estimates for the expected rank of homology, and exhibit explicit nontrivial classes in the nonvanishing regime. These estimates suggest that almost all d-dimensional clique complexes have only one nonvanishing dimension of homology, and we cannot rule out the possibility that they are homotopy equivalent to wedges of a spheres.     

Homology of certain algebras defined by graphs

Let K(G) for a finite graph G with vertices v₁,...,v_n denote the K-algebra with generators X₁,...,X_n and defining relations X_iX_j=X_jX_i if and only if v_i is not connected to v_j by an edge in G. We describe centralizers of monomials, show that the centralizer of a monomial is again a graph algebra, prove a unique factorization theorem for factorizations of monomials into commuting factors, compute the homology of K(G), and show that K(G) is the homology ring of a certain loop space. We also construct a K(π, 1) explicitly where π is the group with generators X₁,...,X_n and defining relations X_iX_j=X_jX_i if and only if v_i is not connected to v_j by an edge in G.     

The evaluation subgroup of a fibre inclusion

Let be a fibration of simply connected CW complexes of finite type with classifying map . We study the evaluation subgroup Gn(E,X;j) of the fibre inclusion as an invariant of the fibre-homotopy type of ξ. For spherical fibrations, we show the evaluation subgroup may be expressed as an extension of the Gottlieb group of the fibre sphere provided the classifying map h induces the trivial map on homotopy groups. We extend this result after rationalization: We show that the decomposition G_∗(E,X;j)⊗Q=(G_∗(X)⊗Q)⊕(π_∗(B)⊗Q) is equivalent to the condition Q(h_?)=0.     

Uniqueness of norm on L(G) and C(G) when G is a compact group

Let G be a compact group. If the trivial representation of G is not weakly contained in the left regular representation of G on L₀²(G) and X is either L^p(G) for 1<p?∞ or C(G), then we show that every complete norm |·| on X that makes translations from (X,|·|) into itself continuous is equivalent to ||·||_p or ||·||_∞ respectively. If 1<p?∞ and every left invariant linear functional on L^p(G) is a constant multiple of the Haar integral, then we show that every complete norm |·| on L^p(G) that makes translations from (L^p(G),|·|) into itself continuous and that makes the map t?L_t from G into bounded is equivalent to ||·||_p.     

A spectral sequence for string cohomology

Let X be a 1-connected space with free-loop space ΛX. We introduce two spectral sequences converging towards H^*(ΛX;Z/p) and H^*((ΛX)_hT;Z/p). The E₂-terms are certain non-Abelian-derived functors applied to H^*(X;Z/p). When H^*(X;Z/p) is a polynomial algebra, the spectral sequences collapse for more or less trivial reasons. If X is a sphere it is a surprising fact that the spectral sequences collapse for p=2.     

Expected rank and randomness in rooted graphs

For a rooted graph G, let EV_b(G;p) be the expected number of vertices reachable from the root when each edge has an independent probability p of operating successfully. We determine the expected value of EV_b(G;p) for random trees, and include a connection to unrooted trees. We also consider rooted digraphs, computing the expected value of a random orientation of a rooted graph G in terms of EV_b(G;p). We consider optimal location of the root vertex for the class of grid graphs, and we also briefly discuss a polynomial that incorporates vertex failure.     

Group-valued continuous functions with the topology of pointwise convergence

Let G be a topological group with the identity element e. Given a space X, we denote by Cp(X,G) the group of all continuous functions from X to G endowed with the topology of pointwise convergence, and we say that X is: (a) G-regular if, for each closed set F⊆X and every point x∈X?F, there exist f∈Cp(X,G) and g∈G?{e} such that f(x)=g and f(F)⊆{e}; (b) G^?-regular provided that there exists g∈G?{e} such that, for each closed set F⊆X and every point x∈X?F, one can find f∈Cp(X,G) with f(x)=g and f(F)⊆{e}. Spaces X and Y are G-equivalent provided that the topological groups Cp(X,G) and Cp(Y,G) are topologically isomorphic.We investigate which topological properties are preserved by G-equivalence, with a special emphasis being placed on characterizing topological properties of X in terms of those of Cp(X,G). Since R-equivalence coincides with l-equivalence, this line of research “includes” major topics of the classical Cp-theory of Arhangel'ski? as a particular case (when G=R).We introduce a new class of TAP groups that contains all groups having no small subgroups (NSS groups). We prove that: (i) for a given NSS group G, a G-regular space X is pseudocompact if and only if Cp(X,G) is TAP, and (ii) for a metrizable NSS group G, a G^?-regular space X is compact if and only if Cp(X,G) is a TAP group of countable tightness. In particular, a Tychonoff space X is pseudocompact (compact) if and only if Cp(X,R) is a TAP group (of countable tightness). Demonstrating the limits of the result in (i), we give an example of a precompact TAP group G and a G-regular countably compact space X such that Cp(X,G) is not TAP.We show that Tychonoff spaces X and Y are T-equivalent if and only if their free precompact Abelian groups are topologically isomorphic, where T stays for the quotient group R/Z. As a corollary, we obtain that T-equivalence implies G-equivalence for every Abelian precompact group G. We establish that T-equivalence preserves the following topological properties: compactness, pseudocompactness, σ-compactness, the property of being a Lindelöf Σ-space, the property of being a compact metrizable space, the (finite) number of connected components, connectedness, total disconnectedness. An example of R-equivalent (that is, l-equivalent) spaces that are not T-equivalent is constructed.     

Characterizing projective general unitary groups PGU<Subscript>3</Subscript>(<Emphasis Type="Italic">q</Emphasis><Superscript>2</Superscript>) by their complex group algebras

Let G be a finite group. Let X ₁(G) be the first column of the ordinary character table of G. We will show that if X ₁(G) = X1(PGU₃(q ²)), then G ? PGU₃(q ²). As a consequence, we show that the projective general unitary groups PGU₃(q ²) are uniquely determined by the structure of their complex group algebras.     

New bounds for the broadcast domination number of a graph

A dominating broadcast on a graph G = (V, E) is a function f: V → {0, 1, ..., diam G} such that f(v) ≤ e(v) (the eccentricity of v) for all v ∈ V and such that each vertex is within distance f(v) from a vertex v with f(v) > 0. The cost of a broadcast f is σ(f) = Σ _v∈V f(v), and the broadcast number λ _b (G) is the minimum cost of a dominating broadcast. A set X ? V(G) is said to be irredundant if each x ∈ X dominates a vertex y that is not dominated by any other vertex in X; possibly y = x. The irredundance number ir (G) is the cardinality of a smallest maximal irredundant set of G. We prove the bound λ_b(G) ≤ 3 ir(G)/2 for any graph G and show that equality is possible for all even values of ir (G). We also consider broadcast domination as an integer programming problem, the dual of which provides a lower bound for λ_b.     

On a characterization of rectangular distributions

Let (X₍₁₎, X₍₂₎) be the order statistics of a sample of size 2 from a population having density ?. It is well known that X₍₁₎ and X₍₂₎ are positively correlated. We show that cov(X₍₁₎, X₍₂₎) has an upper bound which is attained if and only if ? is rectangular density on (0, 1). Our proof uses a 2-dimensional extension of a result due to Polya.     

Labeling graphs with two distance constraints

Given a graph G and integers p,q,d₁ and d₂, with p>q, d₂>d₁?1, an L(d₁,d₂;p,q)-labeling of G is a function f:V(G)→{0,1,2,…,n} such that |f(u)−f(v)|?p if d_G(u,v)?d₁ and |f(u)−f(v)|?q if d_G(u,v)?d₂. A k-L(d₁,d₂;p,q)-labeling is an L(d₁,d₂;p,q)-labeling f such that max_v∈V(G)f(v)?k. The L(d₁,d₂;p,q)-labeling number ofG, denoted by , is the smallest number k such that G has a k-L(d₁,d₂;p,q)-labeling. In this paper, we give upper bounds and lower bounds of the L(d₁,d₂;p,q)-labeling number for general graphs and some special graphs. We also discuss the L(d₁,d₂;p,q)-labeling number of G, when G is a path, a power of a path, or Cartesian product of two paths.     

Some inverse problems involving conditional expectations

Let (Ω, F, P) be a probability space, let H be a sub-σ-algebra of F, and let Y be positive and H-measurable with E[Y] = 1. We discuss the structure of the convex set CE(Y; H) = {X ∈ pF: Y = E[X|H]} of random variables whose conditional expectation given H is the prescribed Y. Several characterizations of extreme points of CE(Y; H) are obtained. A necessary and sufficient condition is given in order that CE(Y; H) be the closed, convex hull of its extreme points. For the case of finite F we explicitly calculate the extreme points of CE(Y; H), identify pairs of adjacent extreme points, and characterize extreme points of CE(Y; H) ? CE(Z; G), where G is a second sub-σ-algebra of F and Z ∈ pG. When H = σ(Y) and appropriate topological hypotheses hold, extreme points of CE(Y; H) are shown to be in explicit one-to-one correspondence with certain left inverses of Y. Finally, it is shown how the same approach can be applied to the problem of extremal random measures on $\text{R}$₊ with a prescribed compensator, to deduce that the number of extreme points is zero or one.     

Extensions and isomorphisms for the generalized Fourier algebras of a locally compact group

It is shown that for amenable groups, all finite-dimensional extensions of A_p(G) algebras split strongly. Furthermore, each extension of A_p(G) which splits algebraically also splits strongly. We also show that if G is an almost connected locally compact group, or a subgroup of GL_n(V) (V being a finite-dimensional vector space), and if for a fixed p∈(1,∞), all finite-dimensional singular extensions of A_p(G) split strongly, then G is amenable. Continuous order isomorphisms for the pointwise order of A_p(G) algebras, are characterized as weighted composition maps. Similarly, order isomorphisms for the pointwise order of B_p(G) algebras, are characterized as ∗-algebra isomorphisms followed by multiplication by an invertible positive multiplier. In addition, it is shown that for amenable groups, an order isomorphism for the pointwise order between A_p(G) algebras that preserve cozero sets is necessarily continuous, and hence induces an algebra isomorphism.     

The global decomposition theorem for Hochschild (co-)homology of singular spaces via the Atiyah-Chern character

We generalize the decomposition theorem of Hochschild, Kostant and Rosenberg for Hochschild (co-)homology to arbitrary morphisms between complex spaces or schemes over a field of characteristic zero. To be precise, we show that for each such morphism X→Y, the Hochschild complex HX/Y, as introduced in [R.-O. Buchweitz, H. Flenner, Global Hochschild (co-)homology of singular spaces, Adv. Math. (2007), doi: 10.1016/j.aim.2007.06.012], decomposes naturally in the derived category D(X) into _⊕p?0Sp(LX/Y[1]), the direct sum of the derived symmetric powers of the shifted cotangent complex, a result due to Quillen in the affine case.Even in the affine case, our proof is new and provides further information. It shows that the decomposition is given explicitly and naturally by the universal Atiyah-Chern character, the exponential of the universal Atiyah class.We further use the decomposition theorem to show that the semiregularity map for perfect complexes factors through Hochschild homology and, in turn, factors the Atiyah-Hochschild character through the characteristic homomorphism from Hochschild cohomology to the graded centre of the derived category.     

Hypergeometric series well-poised in SU(n) and a generalization of Biedenharn's G-functions

We prove that Holman's hypergeometric series well-poised in SU(n) satisfy a general difference equation. We make use of the “path sum” function developed by Biedenharn and this equation to show that a special class of these series, multiplied by simple products, may be regarded as a U(n) generalization of Biedenharn and Louck's G(Δ; X) functions for U(3). The fact that these generalized G-functions are polynomials follows from a detailed study of their symmetries and zeros. As a further application of our general difference equations, we give an elementary proof of Holman's U(n) generalization of the ₅F₄(1) summation theorem.     

Hyper-Tauberian algebras and weak amenability of Figà-Talamanca-Herz algebras

We study certain commutative regular semisimple Banach algebras which we call hyper-Tauberian algebras. We first show that they form a subclass of weakly amenable Tauberian algebras. Then we investigate the basic and hereditary properties of them. Moreover, we show that if A is a hyper-Tauberian algebra, then the linear space of bounded derivations from A into any Banach A-bimodule is reflexive. We apply these results to the Figà-Talamanca-Herz algebra A_p(G) of a locally compact group G for p∈(1,∞). We show that A_p(G) is hyper-Tauberian if the principal component of G is abelian. Finally, by considering the quantization of these results, we show that for any locally compact group G, A_p(G), equipped with an appropriate operator space structure, is a quantized hyper-Tauberian algebra. This, in particular, implies that A_p(G) is operator weakly amenable.