The Realization of 4-dimensional 3-Lie Algebras

In this paper, we mainly investigate the realization of 3-Lie algebras from a family of Lie algebras. We prove the realization theorem, offer a concrete example realizing all type of 4-dimensional 3-Lie algebras, and also give some properties about semi-simple n-Lie algebras.     

4维3-Lie代数的实现

In this paper, we mainly investigate the realization of 3-Lie algebras from a family of Lie algebras. We prove the realization theorem, offer a concrete example realizing all type of 4-dimensional 3-Lie algebras, and also give some properties about semi-simple n-Lie algebras.     

等变上同调与ech超上同调的同构定理

首先证明了任意一个等变微分流形都存在等变良好开覆盖,且等变良好开覆盖集所组成的集合在全部开覆盖组成的集合中共尾.在此基础上,证明了等变上同调与ech超上同调的同构.此定理可应用于实代数簇的Deligne上同调研究.     

L\"obell多面体上的小覆盖

计算了L\"{o}bell多面体上的小覆盖的等变微分同胚类的个数. 在1991年, Davis和Januszkiewicz提出了小覆盖的概念, 给出了组合和拓扑间的一种直接联系, 并证明了单凸多面体上的特征映射($\mathbb{Z}_2^n$染色)与该多面体上的小覆盖一一对应. 文中作者给出了L\"{o}bell多面体上的自同构群和染色规律, 结合Burnside引理计算了一般的L\"{o}bell多面体上的小覆盖的等变微分同胚类的个数.     

单形和3维立方体乘积上小覆盖的等变协边类的个数

本文研究了小覆盖的等变协边分类. 利用示性函数和Stong同态确定了单形和3维立方体乘积上小覆盖的等变协边类的个数, 推广了现有文献中的相关结果.     

单形和3维立方体乘积上小覆盖的等变协边类的个数

本文研究了小覆盖的等变协边分类.利用示性函数和Stong同态确定了单形和3维立方体乘积上小覆盖的等变协边类的个数,推广了现有文献中的相关结果.     

On Fourier coefficients of Maass cusp forms in 3-dimensional hyperbolic space

In this article we establish the analogue of a theorem of Kuznetsov (theorem 6 of [3]) in the case of 3-dimensional hyperbolic space. We also consider a generalization of this result for higher dimensional hyperbolic spaces and discuss the relevant ingredients of a proof. Dedicated to the memory of Professor K G Ramanathan     

A van Kampen theorem for equivariant fundamental groupoids

The equivariant fundamental groupoid of a G-space X is a category which generalizes the fundamental groupoid of a space to the equivariant setting. In this paper, we prove a van Kampen theorem for these categories: the equivariant fundamental groupoid of X can be obtained as a pushout of the categories associated to two open G-subsets covering X. This is proved by interpreting the equivariant fundamental groupoid as a Grothendieck semidirect product construction, and combining general properties of this construction with the ordinary (non-equivariant) van Kampen theorem. We then illustrate applications of this theorem by showing that the equivariant fundamental groupoid of a G-CW complex only depends on the 2-skeleton and also by using the theorem to compute an example.     

棱柱上的小覆盖的等变配边分类

多面体上的小覆盖的等变配边类是由它的切表示集所决定的.本文通过将棱柱上的小覆盖的切表示集约化到一种素形式,来确定其等变配边分类.     

Operations in A-theory

A construction for Segal operations for K-theory of categories with cofibrations, weak equivalences and a biexact pairing is given and coherence properties of the operations are studied. The model for K-theory, which is used, allows coherence to be studied by means of (symmetric) monoidal functors. In the case of Waldhausen A-theory it is shown how to recover the operations used in Waldhausen (Lecture Notes in Mathematics, Vol. 967, Springer, Berlin, 1982, pp. 390-409) for the A-theory Kahn-Priddy theorem. The total Segal operation for A-theory, which assembles exterior power operations, is shown to carry a natural infinite loop map structure. The basic input is the un-delooped model for K-theory, which has been developed from a construction by Grayson and Gillet for exact categories in Gunnarsson et al. (J. Pure Appl. Algebra 79 (1992) 255), and Grayson's setup for operations in Grayson (K-theory (1989) 247). The relevant material from these sources is recollected followed by observations on equivariant objects and pairings. Grayson's conditions are then translated to the context of categories with cofibrations and weak equivalences. The power operations are shown to be well behaved w.r.t. suspension and are extended to algebraic K-theory of spaces. Staying close with the philosophy of Waldhausen (1982) Waldhausen's maps are found. The Kahn-Priddy theorem follows from splitting the “free part” off the equivariant theory. The treatment of coherence of the total operation in A-theory involves results from Laplaza (Lecture Notes in Mathematics, Vol. 281, Springer, Berlin, 1972, pp. 29-65) and restriction to spherical objects in the source of the operation.     

Multiplicative structure in equivariant cohomology

We introduce the notion of a strongly homotopy-comultiplicative resolution of a module coalgebra over a chain Hopf algebra, which we apply to proving a comultiplicative enrichment of a well-known theorem of Moore concerning the homology of quotient spaces of group actions. The importance of our enriched version of Moore’s theorem lies in its application to the construction of useful cochain algebra models for computing multiplicative structure in equivariant cohomology.In the special cases of homotopy orbits of circle actions on spaces and of group actions on simplicial sets, we obtain small, explicit cochain algebra models that we describe in detail.     

循环多胞形$C^{3}(6)$的对偶和单形乘积上的小覆盖

计算了循环多胞形$C^{3}(6)$的对偶和单形乘积上(可定向)小覆盖的等变同胚类的个数     

Equivariant Chow groups for torus actions

We study Edidin and Graham's equivariant Chow groups in the case of torus actions. Our main results are: (i) a presentation of equivariant Chow groups in terms of invariant cycles, which shows how to recover usual Chow groups from equivariant ones; (ii) a precise form of the localization theorem for torus actions on projective, nonsingular varieties; (iii) a construction of equivariant multiplicities, as functionals on equivariant Chow groups; (iv) a construction of the action of operators of divided differences on theT-equivariant Chow group of any scheme with an action of a reductive group with maximal torusT. We apply these results to intersection theory on varieties with group actions, especially to Schubert calculus and its generalizations. In particular, we obtain a presentation of the Chow ring of any smooth, projective spherical variety.     

Convexity of Multi-valued Momentum Maps

A famous theorem of Atiyah, Guillemin and Sternberg states that, given a Hamiltonian torus action, the image of the momentum map is a convex polytope. We prove that this result can be extended to the case in which the action is non-Hamiltonian. Our generalization of the theorem states that, given a symplectic torus action, the momentum map can be defined on an appropriate covering of the manifold and its image is the product of a convex polytope along a rational subspace times the orthogonal vector space. We also prove that this decomposition in direct product is stable under small equivariant perturbations of the symplectic structure; this, in particular, means that the property of being Hamiltonian is locally stable. The technique developed allows us to extend the result to any compact group action and also to deduce that any symplectic n-torus action, with fixed points, on a compact 2n-dimensional manifold, is Hamiltonian.     

Brauer induction and equivariant stable homotopy

Brauer's induction theorem, published in 1951, asserts that every element of the complex representation ring R(G) of a finite group G is a linear combination of classes induced from 1-dimensional representations of subgroups of G. In 1987, Snaith formulated an explicit version of the induction theorem. Using the methods of equivariant fibrewise stable homotopy theory, specifically fixed-point theory, this note clarifies the relation between the explicit Brauer induction theorem due to Snaith, Boltje and Symonds and a topological splitting theorem established by Segal in 1973.     

Small covers over prisms

In this paper we calculate the number of equivariant diffeomorphism classes of small covers over a prism.     

Cohomological rigidity and the number of homeomorphism types for small covers over prisms

In this paper, based upon the basic theory for glued manifolds in M.W. Hirsch (1976) [8, Chapter 8, §2 Gluing Manifolds Together], we give a method of constructing homeomorphisms between two small covers over simple convex polytopes. As a result we classify, up to homeomorphism, all small covers over a 3-dimensional prism P³(m) with m?3. We introduce two invariants from colored prisms and other two invariants from ordinary cohomology rings with Z₂-coefficients of small covers. These invariants can form a complete invariant system of homeomorphism types of all small covers over a prism in most cases. Then we show that the cohomological rigidity holds for all small covers over a prism P³(m) (i.e., cohomology rings with Z₂-coefficients of all small covers over a P³(m) determine their homeomorphism types). In addition, we also calculate the number of homeomorphism types of all small covers over P³(m).     

Simple Cubature Formulas with High Polynomial Exactness

We study cubature formulas for d -dimensional integrals with arbitrary weight function of tensor product form. We present a construction that yields a high polynomial exactness: for fixed degree, the number of knots depends on the dimension in an order-optimal way. The cubature formulas are universal: the order of convergence is almost optimal for two different scales of function spaces. The construction is simple: a small number of arithmetical operations is sufficient to compute the knots and the weights of the formulas. August 25, 1997. Date revised: December 3, 1998. Date accepted: March 3, 1999.     

A General Approach to Equivariant Biharmonic Maps

In this paper we describe a 1-dimensional variational approach to the analytical construction of equivariant biharmonic maps. Our goal is to provide a direct method which enables analysts to compute directly the analytical conditions which guarantee biharmonicity in the presence of suitable symmetries. In the second part of our work, we illustrate and discuss some examples. In particular, we obtain a 1-dimensional stability result, and also show that biharmonic maps do not satisfy the classical maximum principle proved by Sampson for harmonic maps.     

Symmetries in Projective Multiresolution Analyses

We give an equivariant version of Packer and Rieffel’s theorem on sufficient conditions for the existence of orthonormal wavelets in projective multiresolution analysis. Suppose that the scaling functions are invariant with respect to some finite group action. We give sufficient conditions for the existence of wavelets with similar invariance. Research supported in part by the Research Council of Norway, project number NFR 154077/420. Some of the final work was also done with the support from the project NFR 170620/V30.