The equivariant Serre spectral sequence as an application of a spectral sequence of Spanier

E.H. Spanier (1992) has constructed, for a cohomology theory defined on a triangulated space and locally constant on each open simplex, a spectral sequence whose E₂-term consists of certain simplicial cohomology groups, converging to the cohomology of the space. In this paper we study a closed G-fibration ƒ: Y → X, where G is a finite group. We show that if the base-G-spaceX is equivariantly triangulated and Y is paracompact, then Spanier's spectral sequence yields an equivariant Serre spectral sequence for ƒ. The main point here is to identify the equivariant singular cohomology groups of X with appropriate simplicial cohomology groups of the orbit space X/G.     

On deformations of transversely homogeneous foliations

A transversely homogeneous foliation is a foliation whose transverse model is a homogeneous space G/H. In this paper we consider the class of transversely homogeneous foliations on a manifold M which can be defined by a family of 1-forms on M fulfilling the Maurer–Cartan equation of the Lie group G. This class includes as particular cases Lie foliations and certain homogeneous spaces foliated by points. We develop, for the foliations belonging to this class, a deformation theory for which both the foliation and the model homogeneous space G/H are allowed to change. As the main result we show that, under some cohomological assumptions, there exist a versal space of deformations of finite dimension for the foliations of the class and when the manifold M is compact. Some concrete examples are discussed.     

Bredon-Illman cohomology with actions of totally disconnected groups

We study the Bredon-Illman cohomology with local coefficients for a G-space X in the case of G being a totally disconnected, locally compact group. We prove that any short exact sequence of equivariant local coefficients systems on X gives a long exact sequence of the associated Bredon-Illman cohomology groups with local coefficients.     

On Some Families of Smooth Affine Spherical Varieties of Full Rank

Let G be a complex connected reductive group. Losev has shown that a smooth affine spherical G-variety X is uniquely determined by its weight monoid, which is the set of irreducible representations of G that occur in the coordinate ring of X. In this paper we use a combinatorial characterization of the weight monoids of smooth affine spherical varieties to classify:(a) all such varieties for G = SL(2) × C~×and(b) all such varieties for G simple which have a G-saturated weight monoid of full rank. We also use the characterization and Knop's classification theorem for multiplicity free Hamiltonian manifolds to give a new proof of Woodward's result that every reflective Delzant polytope is the moment polytope of such a manifold.     

On the structure of the group of equivariant diffeomorphisms of G-manifolds with codimension one orbit

It is known that the equivariant diffeomorphism group Diff_G^∞(M)₀ of a principal G-manifold M is perfect. If M has at least two orbit types, then it is not true. The purpose of this paper is to determine the first homology group of Diff_G^∞(M)₀ when M is a G-manifold with codimension one orbit.     

The equivariant Euler characteristic

We describe an equivariant version of the Euler characteristic in order to extend to the equivariant case classical results relating the Euler characteristic to vector field (Reinhart) bordism of smooth manifolds and controllable cut-and-paste equivalence. We show that the nonequivariant results continue to hold for an arbitrary finite ambient group G, both in the oriented and unoriented cases, and thereby extend work on this subject begun by several authors. We use a new definition of equivariant orientation in terms of a categorical notion of ‘groupoid representations’.     

Numerical linear algorithms and group theory

Most methods for solving linear systems Ax=b are founded on the ability to split up the matrix A in the form of a product A=G·R with G belonging to a subgroup of the general linear group Gl(n,R) and R being a regular upper triangular matrix. In the same way, the calculation of the eigenvalues of a matrix by the use of an algorithm of the Rutishauser type is based on a G·R decomposition for the matrix. Our aim in this article will be to show the importance of the notion of splitting up, to set out the conditions under which it may be used and to show how it enables us to generate new algorithms.     

Reconstructing finite group actions and characters from subgroup information

A finite group G is said to be action reconstructible if, for any action of G on a finite set, the numbers of orbits under restriction to each subgroup always give enough information to reconstruct the action up to equivalence. G is character reconstructible if, given any matrix representation of G, the mean value of the character on each subgroup always gives enough information to reconstruct the character. The conjugacy matrix of G is the matrix whose (ij) entry is the number of elements of the jth conjugacy class belonging to a typical subgroup of the ith subgroup conjugacy class. It is shown that G is action reconstructible if and only if the rows of this matrix are linearly independent (which is in turn true if and only if G is cyclic), and is character reconstructible if and only if the columns are linearly independent (which is true if and only if any two elements of G which generate conjugate cyclic subgroups are themselves conjugate).     

Solvable block transitive automorphism groups of 2−(v,5,1) designs

Let G be a solvable block transitive automorphism group of a 2−(v,5,1) design and suppose that G is not flag transitive. We will prove that

(1) if G is point imprimitive, then v=21, and GZ₂₁:Z₆;

(2) if G is point primitive, then GAΓL(1,v) and v=p^a, where p is a prime number with p≡21 (mod 40), and a an odd integer.

     

The exponent of the primitive Cayley digraphs on finite Abelian groups

Let G be finite group and let S be a subset of G. We prove a necessary and sufficient condition for the Cayley digraph X_{(G, S)} to be primitive when S contains the central elements of G. As an immediate consequence we obtain that a Cayley digraph X_{(G, S)} on an Abelian group is primitive if and only if S^−1S is a generating set for G. Moreover, it is shown that if a Cayley digraph X_{(G, S)} on an Abelian group is primitive, then its exponent either is or is not exceeding . Finally, we also characterize those Cayley digraphs on Abelian groups with exponent . In particular, we generalize a number of well-known results for the primitive circulant matrices.     

List colourings of planar graphs

A graph G = G(V, E) is called L-list colourable if there is a vertex colouring of G in which the colour assigned to a vertex v is chosen from a list L(v) associated with this vertex. We say G is k-choosable if all lists L(v) have the cardinality k and G is L-list colourable for all possible assignments of such lists. There are two classical conjectures from Erd s, Rubin and Taylor 1979 about the choosability of planar graphs:

every planar graph is 5-choosable and,

there are planar graphs which are not 4-choosable.

We will prove the second conjecture.     

Normal Sylow subgroups and monomial Brauer characters

Let G be a finite group, p be a prime divisor of |G|, and P be a Sylow p-subgroup of G. We prove that P is normal in a solvable group G if |G : ker φ|p' = φ(1)p' for every nonlinear irreducible monomial p-Brauer character φ of G, where ker φ is the kernel of φ and φ(1)p' is the p'-part of φ(1).     

On the exponential map of borei subgroups in a semi-simple lie group

Let G be a connected, complex, semi-simple Lie group Let g be an element in G. Let B be a Borel subgroup of G and g in B. Let m and n be the least positive integers such that the element g^m lies on a one-parameter subgroup in G and the element gⁿ lies on a one-parameter subgroup in B. We denote these integers by ind_G(g) and ind_B(g). In this note we prove the conjecture ind_G(g) = ind_B(g), if g is regular.     

Symmetry classes of tensors:dual indices

In this article we use the concept of r-depth to introduce the notion of dual multilinearity index. Using this concept we associate to each irreducible character of a subgroup G of S_m an improper partition of m. It is proved that of G = S_m this partition is the partition usually associated to the irreducible characters of S_m.     

On index of total boundedness of (strictly) o-bounded groups

Let G be a metrizable topological group. Denote by itb(G) the smallest cardinality of a cover of G by totally bounded subsets of G. A group G is defined to be σ-bounded if itb(G)₀. The group G is called o-bounded if for every sequence (U_n)_nω of neighborhoods of the identity in G there exists a sequence (F_n)_nω of finite subsets in G such that G=_nωF_n·U_n; G is called strictly o-bounded (respectively OF-determined) if the second player (respectively one of the players) has a winning strategy in the following game OF: two players, I and II, choose at every step n an open neighborhood U_n of the identity in G and a finite subset F_n of G, respectively. The player II wins if G=_nωF_n·U_n.

For a second countable group G the following results are proven. . If G is strictly o-bounded, then itb(G)₁ and G is σ-bounded or meager. If the space G is analytic, then the group is OF-determined and satisfies . G is σ-bounded if it is strictly o-bounded and one of the following conditions holds: (i) G is analytic; (ii) ; (iii) (MA+¬CH) holds; (iv) analytic games are determined; (v) there exists a measurable cardinal. Also we show that under (MA) every non-locally compact Polish Abelian divisible group contains a Baire o-bounded OF-undetermined subgroup.     

On <Emphasis Type="Italic">σ</Emphasis>-semipermutable Subgroups of Finite Groups

Let σ={σ_i|i∈I} be some partition of the set of all primes P, G a finite group and σ(G)={σ_i|σ_i ∩ π(G)≠∅}. A set H of subgroups of G is said to be a complete Hall σ-set of G if every member≠1 of H is a Hall σ_i-subgroup of G for some σ_i ∈ σ and H contains exactly one Hall σ_i-subgroup of G for every σ_i ∈ σ(G). A subgroup H of G is said to be:σ-semipermutable in G with respect to H if HH_i^x=H_i^xH for all x ∈ G and all H_i ∈ H such that (|H|,|H_i|)=1; σ-semipermutable in G if H is σ-semipermutable in G with respect to some complete Hall σ-set of G. We study the structure of G being based on the assumption that some subgroups of G are σ-semipermutable in G.     

Lower bounds of the Laplacian spectrum of graphs based on diameter

Let G be a connected graph of order n. The diameter of G is the maximum distance between any two vertices of G. In the paper, we will give some lower bounds for the algebraic connectivity and the Laplacian spectral radius of G in terms of the diameter of G.     

Connectivity of generalized prisms over G

The problem of building larger graphs with a given graph as an induced subgraph is one which can arise in various applications and in particular can be important when constructing large communications networks from smaller ones. What one can conclude from this paper is that generalized prisms over G may provide an important such construction because the connectivity of the newly created graph is larger than that of the original (connected) graph, regardless of the permutation used.

For a graph G with vertices labeled 1,2,…, n and a permutation in S_n, the generalized prisms over G, (G) (also called a permutation graph), consists of two copies of G, say G_x and G_y, along with the edges (x_i, y_(i), for 1≤i≤n. The purpose of this paper is to examine the connectivity of generalized prisms over G. In particular, upper and lower bounds are found. Also, the connectivity and edge-connectivity are determined for generalized prisms over trees, cycles, wheels, n-cubes, complete graphs, and complete bipartite graphs. Finally, the connectivity of the generalized prism over G, (G), is determined for all in the automorphism group of G.     

Nonsolvable groups whose irreducible character degrees have special 2-parts

Let G be a nonsolvable group and Irr(G) the set of irreducible complex characters of G. We consider the nonsolvable groups whose character degrees have special 2-parts and prove that if χ(1)₂ = 1 or |G|₂ for every χ ∈ Irr(G), then there exists a minimal normal subgroup N of G such that N ≅ PSL(2, 2ⁿ) and G/N is an odd order group.