M?bius transform, moment-angle complexes and Halperin–Carlsson conjecture

The motivation for this paper comes from the Halperin–Carlsson conjecture for (real) moment-angle complexes. We first give an algebraic combinatorics formula for the M?bius transform of an abstract simplicial complex K on [m]={1,…,m} in terms of the Betti numbers of the Stanley–Reisner face ring k(K) of K over a field k. We then employ a way of compressing K to provide the lower bound on the sum of those Betti numbers using our formula. Next we consider a class of generalized moment-angle complexes Z_K^{(\mathbb D, \mathbb S)}\mathcal{Z}_{K}^{(\underline{\mathbb{ D}}, \underline{\mathbb{ S}})}, including the moment-angle complex Z_K\mathcal{Z}_{K} and the real moment-angle complex \mathbbRZ_K\mathbb{R}\mathcal {Z}_{K} as special examples. We show that H^*(Z_K^{(\mathbb D, \mathbb S)};k)H^{*}(\mathcal{Z}_{K}^{(\underline{\mathbb{ D}}, \underline{\mathbb{ S}})};\mathbf{k}) has the same graded k-module structure as Tor  ^k[v](k(K),k). Finally we show that the Halperin–Carlsson conjecture holds for Z_K\mathcal{Z}_{K} (resp. \mathbb RZ_K\mathbb{ R}\mathcal{Z}_{K}) under the restriction of the natural T ^m -action on Z_K\mathcal{Z}_{K} (resp. (ℤ₂) ^m -action on \mathbb RZ_K\mathbb{ R}\mathcal{Z}_{K}).     

On theG-matrices with entries and eigenvalues inQ(i)

LetG be a finite abelian group,K a subfield ofC, C[G] regarded as an algebra of matrices.A _G ^K {AC[G]| all the entries and eigenvalues ofA are inK} is an association algebra overK. In this paper, the association scheme ofA _G ^K is determined and in the caseK=Q(i), the first eigenmatrix of the association scheme computed. As an application, it is proved thatZ ₄×Z ₄×Z ₄ is the only abelian group admitted as a Singer group by some distance-regular digraph of girth 4 on 64 vertices.     

Higher Chern Classes and Steenrod Operations in Motivic Cohomology

In this short paper we investigate the relation between higher Chern classes and reduced power operations in motivic cohomology. More precisely, we translate the well-known arguments [5] into the context of motivic cohomology and define higher Chern classes c_p,q : K _p(X) → H^2q-p (X,Z(q)) → H^2q-p(X, Z/l(q)), where X is a smooth scheme over the base field k, l is a prime number and char(k) ≠ l. The same approach produces the classes for K-theory with coefficients as well. Let further Pⁱ : H^m(X, Z/l(n)) → H^m+2i(l-1) (X, Z/l(n + i(l - 1))) denote the ith reduced power operation in motivic cohomology, constructed in [2]. The main result of the paper looks as follows.     

Moment-angle complexes from simplicial posets

We extend the construction of moment-angle complexes to simplicial posets by associating a certain T ^m -space Z _S to an arbitrary simplicial poset S on m vertices. Face rings ℤ[S] of simplicial posets generalise those of simplicial complexes, and give rise to new classes of Gorenstein and Cohen-Macaulay rings. Our primary motivation is to study the face rings ℤ[S] by topological methods. The space Z _S has many important topological properties of the original moment-angle complex Z _K associated to a simplicial complex K. In particular, we prove that the integral cohomology algebra of Z _S is isomorphic to the Tor-algebra of the face ring ℤ[S]. This leads directly to a generalisation of Hochster’s theorem, expressing the algebraic Betti numbers of the ring ℤ[S] in terms of the homology of full subposets in S. Finally, we estimate the total amount of homology of Z _S from below by proving the toral rank conjecture for the moment-angle complexes Z _S .     

On nonnegatively curved 4-manifolds with discrete symmetry

Let M be the closed, simply connected, 4-manifold with nonnegative sectional curvature, called a nonnegatively curved 4-manifold, with an effective and isometric Z _m -action for a positive integer m ≧ 61⁷. Assume that Z _m acts trivially on the homology of M. The goal of this short paper is to prove that if the fixed point set of any nontrivial element of Z _m has at most one two-dimensional component, then M is homeomorphic to S ⁴, # _i ^l =1S ² × S ², l = 1, 2, or # _j ^k = 1 ± CP ², k = 1, 2, 3, 4, 5. The main strategy of this paper is to give an upper bound of the Euler characteristic χ(M) under the homological assumption of a Z _m -action as above by using the Lefschetz fixed point formula.     

Initial layers of Zp-extensions and Greenberg's conjecture

Let p be a prime number, K a finite abelian extension of Q containing p-th roots of unity and K _n the n-th layer of the cyclotomic Z _p -extension of K. Under some conditions we construct an element of K _n from an ideal class of the maximal real subfield of K _n . We determine whether its p-th root is contained by some Z _p -extension of K _n or not for each n, using the zero of p-adic L-function and the order of the ideal class group of the maximal real subfield of K _m for sufficiently large m. Received: 13 February 1998 / Revised version: 30 September 1998     

Vandermonde Matrices, NP-Completeness, and Transversal Subspaces

Let K be an infinite field. We give polynomial time constructions of families of r-dimensional subspaces of K ⁿ with the following transversality property: any linear subspace of K ⁿ of dimension n–r is transversal to at least one element of the family. We also give a new NP-completeness proof for the following problem: given two integers n and m with n \leq m and a n × m matrix A with entries in Z, decide whether there exists an n × n subdeterminant of A which is equal to zero.     

Self-dual codes over the Kleinian four group

We introduce self-dual codes over the Kleinian four group K=Z ₂×Z ₂ for a natural quadratic form on K ⁿ and develop the theory. Topics studied are: weight enumerators, mass formulas, classification up to length 8, neighbourhood graphs, extremal codes, shadows, generalized t-designs, lexicographic codes, the Hexacode and its odd and shorter cousin, automorphism groups, marked codes. Kleinian codes form a new and natural fourth step in a series of analogies between binary codes, lattices and vertex operator algebras. This analogy will be emphasized and explained in detail.     

Singular integrals associated to the Laplacian on the affine groupax+b

We consider singular integral operators of the form (a)Z ₁L⁻¹Z₂, (b)Z ₁Z₂L⁻¹, and (c)L ⁻¹Z₁Z₂, whereZ ₁ andZ ₂ are nonzero right-invariant vector fields, andL is theL ²-closure of a canonical Laplacian. The operators (a) are shown to be bounded onL ^p for allp∈(1, ∞) and of weak type (1, 1), whereas all of the operators in (b) and (c) are not of weak type (p, p) for anyp∈[1, ∞). Research supported by the Australian Research Council. Research carried out as a National Research Fellow.     

Fortsetzung Ganzzahliger Cozyklen von Kompakten Teilen

Let Y be a topological space and X a subspace of Y. We assume that X is the union of an increasing sequence of subspaces K_S such that every quasi-compact subset of X is contained in some K_S and the singular homology groups of all K_S are finitely generated. The object of this paper is to give a purely algebraic characterisation of the following subgroup of H^q (X,Z): it consists of all those elements in H^q (X,Z) whose image in each of the H^q (K_S,Z) lies in the image of the induced homomorphism H^q (Y,Z)H^q (K_S,Z), These subgroups are encountered in Runge approximation theory. Partial results were obtained in an earlier common paper with K. Stein, [1].

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Meinem verehrten Lehrer Karl Stein zum 60. Geburtstag gewidmet     

On the Erd s-Ginzburg-Ziv theorem and the Ramsey numbers for stars and matchings

A link between Ramsey numbers for stars and matchings and the Erd s-Ginzburg-Ziv theorem is established. Known results are generalized. Among other results we prove the following two theorems. Theorem 5. Let m be an even integer. If c : e (K_2m−1)→{0, 1,…, m−1} is a mapping of the edges of the complete graph on 2m−1 vertices into {0, 1,…, m−1}, then there exists a star K_1,m in K_2m−1 with edges e₁, e₂,…, e_m such that c(e₁)+c(e₂)++c(e_m)≡0 (mod m). Theorem 8. Let m be an integer. If c : e(K^r_(r+1)m−1)→{0, 1,…, m−1} is a mapping of all the r-subsets of an (r+1)m−1 element set S into {0, 1,…, m−1}, then there are m pairwise disjoint r-subsets Z₁, Z₂,…, Z_m of S such that c(Z₁)+c(Z₂)++c(Z_m)≡0 (mod m).     

The plancherel measure for symmetric graphs

Summary Let G be the free product of r copies of the cyclic group Z _k.We obtain the Plancherel formula for the commutative O ^*-algebra of radial convolution operators on l ² (G). The Plancherel measure is expressed in terms of the c-function appearing in the expansion of spherical functions on G as linear combinations of exponentials.     

A characterization ofC 1-convex sets in Sobolev spaces

In this paper we prove that, ifK is a closed subset ofW ₀ ^1,p (Ω,R ^m ) with 1<p<+∞ andm≥1, thenK is stable under convex combinations withC ¹ coefficients if and only if there exists a closed and convex valued multifunction from Ω toR ^m such that The casem=1 has already been studied by using truncation arguments which rely on the order structure ofR (see [2]). In the casem>1 a different approach is needed, and new techniques involving suitable Lipschitz projections onto convex sets are developed. Our results are used to prove the stability, with respect to the convergence in the sense of Mosco, of the class of convex sets of the form (*); this property may be useful in the study of the limit behaviour of a sequence of variational problems of obstacle type. This article was processed by the author using the Springer-Verlag TEX mamath macro package 1990     

Multiplicative Semiprimeness of Skew Lie Algebras

Abstract

Let A be a semiprime associative algebra with an involution over a field of characteristic not 2, let K be the Lie algebra of all skew elements of A, and let Z _{[K, K]} denote the annihilator of the Lie algebra [K, K]. We will prove that the multiplication algebra of the semiprime Lie algebra [K, K]/Z _{[K, K]} is also semiprime. As a consequence, the multiplication algebra of [K, K]/Z _{[K, K]} is prime, whenever [K, K]/Z _{[K, K]} is prime. We will obtain similar results for the Lie algebra K/Z _K whenever the base field has characteristic zero.     

The structure of symmetry groups of GK(2n,G)

The automorphism groups of the one-factorizations GK(2n,G) are computed. It is shown that every 1-factorization of K_2n with a subgroup of the automorphism group that acts sharply 2-transitively on the one-factors must be GK(p^m + 1, (Z_p)^m) for some odd prime p. © 1994 John Wiley & Sons, Inc.     

The degree of an integral point in Ps minus a hypersurface

Let K be a number field of degree m with ring of integers R and absolute discriminant d_K. Given a hypersurface Z_K of degree d in the projective space P_K^us over K with Zariski closure Z in P_R^s, we give an explicit function of m, d_K, s,d, a Hermitian metric on R^s+1 ⊗_z C, and a projective height of Z defined in [1], 4.1, such that there exists an integral point in P_R^s Z of degree bounded by this function.     

Generalized composition and Volterra type operators between QK spaces

Abstract

In this paper, the boundedness and compactness of the generalized composition operators and the products of Volterra type operators a nd composition operators between Q_K spaces are investigated. We also give a necessary condition for multiplication operators between Q_K spaces to be bounded or compact.     

Grassmannians of 2-Sided Vector Spaces

Let k ? K be an extension of fields, and let A ? M _n (K) be a k-algebra. We study parameter spaces of m-dimensional subspaces of K ⁿ which are invariant under A. The space  _A (m, n), whose R-rational points are A-invariant, free rank m summands of R ⁿ , is well known. We construct a distinct parameter space,  _A (m, n), which is a fiber product of a Grassmannian and the projectivization of a vector space. We then study the intersection  _A (m, n) ∩  _A (m, n), which we denote by  _A (m, n). Under suitable hypotheses on A, we construct affine open subschemes of  _A (m, n) and  _A (m, n) which cover their K-rational points. We conclude by using  _A (m, n),  _A (m, n), and  _A (m, n) to construct parameter spaces of 2-sided subspaces of 2-sided vector spaces.     

Singular integrals on the discrete Besov space B1 0,1 (Z)

Summary We show the boundedness of singular integral operators on the discrete Besov space B₁^0,1 (Z). For this purpose, we introduce discrete special molecules on B₁^0,1 (Z).     

Pseudodifference Operators on Weighted Spaces, and Applications to Discrete Schrödinger Operators

We study pseudodifference operators on Z ^N with symbols which are bounded on Z ^N ×T ^N together with their derivatives with respect to the second variable. In the same way as partial differential operators on R ^N are included in an algebra of pseudodifferential operators, difference operators on Z ^N are included in an algebra of pseudodifference operators. Particular attention is paid to the Fredholm properties of pseudodifference operators on general exponentially weighted spaces l _w ^p (Z ^N ) and to Phragmen–Lindelöf type theorems on the exponential decay at infinity of solutions to pseudodifference equations. The results are applied to describe the essential spectrum of discrete Schrödinger operators and the decay of their eigenfunctions at infinity.