Non-Archimedean integrals and stringy Euler numbers of log-terminal pairs

Using non-Archimedian integration over spaces of arcs of algebraic varieties, we define stringy Euler numbers associated with arbitrary Kawamata log-terminal pairs. There is a natural Kawamata log-terminal pair corresponding to an algebraic variety V having a regular action of a finite group G. In this situation we show that the stringy Euler number of this pair coincides with the physicists’ orbifold Euler number defined by the Dixon-Harvey-Vafa-Witten formula. As an application, we prove a conjecture of Miles Reid on the Euler numbers of crepant desingularizations of Gorenstein quotient singularities. Received March 19, 1998     

A counterexample to questions on the integrality property of virtual signature

Wall (1961), defined the virtual Euler Characteristic χ(Γ) of an arbitrary group Γ of finite homological type as , where Γ′ is any torsion free subgroup of finite index in Γ. Analogous to virtual Euler Characteristic, we define the Virtual signature of an oriented virtual Poincare Duality group, a rational number. We show that two of Ken Brown's results on questions regarding the integrality property of virtual Euler Characteristics when formulated in the Virtual signature case is false.     

About Knop's Action of the Weyl Group on the Set of Orbits of a Spherical Subgroup in the Flag Manifold

Let G be a connected reductive algebraic group defined on an algebraically closed field k of characteristic different from 2. Let B denote the flag variety of G. Let H be a spherical subgroup of G. F. Knop defined an action of the Weyl group W of G on the finite set of the H-orbits in B. Here, we define an invariant, namely the type, separating the orbits of W.     

Hereditary noetherian categories of positive Euler characteristic

We develop the fundamentals of hereditary noetherian categories with Serre duality over an arbitrary field k, where the category of coherent sheaves over a smooth projective curve over k serves as the prime example and others are coming from the representation theory of finite dimensional algebras. The proper way to view such a category is to think of coherent sheaves on a possibly non-commutative smooth projective curve. We define for each such category notions like function field and Euler characteristic, determine its Auslander-Reiten components and study stable and semistable bundles for an appropriate notion of degree. We provide a complete classification of hereditary noetherian categories for the case of positive Euler characteristic by relating these to finite dimensional representations of (locally bounded) hereditary k-algebras whose underlying valued quiver admits a positive additive function. Dedicated to Otto Kerner on the occasion of his 60th birthday     

Orbifold Euler characteristics and the number of commutingm-tuples in the symmetric groups

Generating functions for the number of commutingm-tuples in the symmetric groups are obtained. We define a natural sequence of orbifold Euler characteristics for a finite groupG acting on a manifoldX. Our definition generalizes the ordinary Euler characteristic ofX/G and the string-theoretic orbifold Euler characteristic. Our formulae for commutingm-tuples underlie formulae that generalize the results of Macdonald and Hirzebruch-Höfer concerning the ordinary and string-theoretic Euler characteristics of symmetric products.Supported partly by a grant from the Ford Foundation.     

Cohomogeneity one manifolds with positive Euler characteristic

We classify those manifolds of positive Euler characteristic on which a Lie group G acts with cohomogeneity one, where G is classical simple.     

Rings associated with hyperbolic groups

Abstract We define quasiconvexity cone Qcone(τ) over an infinite hyperbolic (in the sense of Gromov) group τ as the set of conjugacy classes of infinite quasiconvex subgroups H?τ and show that the abelian group of Qcone(τ)-divisors, i.e. finite sums of points from Qcone(τ) with integer coefficients, can be equipped with a natural structure of commutative associative ring with identity. Euler characteristic can be considered as a rational-valued function on Qcone(τ). This approach gives another point of view on the strengthened form of Hanna Neumann's conjecture on the maximal rank of the intersection of two finitely generated subgroups of the free group on two generators.     

The normalized cyclomatic quotient associated with presentations of finitely generated groups

Given a presentation of ann-generated group, we define the normalized cyclomatic quotient (NCQ) of it, which gives a number between 1−n and 1. It is computed through an investigation of the asymptotic behavior of a kind of an “average rank”, or more precisely the quotient of the rank of the fundamental group of a finite subgraph of the corresponding Cayley graph by the size of the subgraph. In many ways (but not always) the NCQ behaves similarly to the behavior of the spectral radius of a symmetric random walk on the graph. In particular, it characterizes amenable groups. For some types of groups, like finite, amenable or free groups, its value equals that of the Euler characteristic of the group. We give bounds for the value of the NCQ for factor groups and subgroups, and formulas for its value on direct and free products. Some other asymptotic invariants are also discussed.     

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’.     

The Euler characteristic of a polyhedral product

Given a finite simplicial complex L and a collection of pairs of spaces indexed by the vertices of L, one can define the ??polyhedral product?? of the collection with respect to L. We record a simple formula for its Euler characteristic. In special cases the formula simplifies further to one involving the h-polynomial of L.     

Normal Abelian subgroups and Euler characteristic

We generalize Rosset's theorem which states that the Euler characteristic of a group G of type FL_C vanishes if G contains a torsion free normal subgroup. In our case the subgroup is allowed to have torsion (but must also have elements of infinite order). Under similar conditions on the regular covering of a finite CW-complex X, it is shown that the Euler characteristic of X is 0; this includes the special case where X is nilpotent.     

Euler Characteristics on virtually free products

We define Euler characteristics on classes of residually finite and virtually torsion free groups and we show that they satisfy certain formulas in the case of amalgamated free products and HNN extensions over finite subgroups. These formulas are obtained from a general result which applies to the rank gradient and the first L²?Betti number of a finitely generated group.     

The euler characteristic is the unique locally determined numerical homotopy invariant of finite complexes

If a numerical homotopy invariant of finite simplicial complexes has a local formula, then, up to multiplication by an obvious constant, the invariant is the Euler characteristic. Moreover, the Euler characteristic itself has a unique local formula.     

The Angle Defect for Odd-Dimensional Simplicial Manifolds

In a 1967 paper, Banchoff stated that a certain type of polyhedral curvature, that applies to all finite polyhedra, was zero at all vertices of an odd-dimensional polyhedral manifold; one then obtains an elementary proof that odd-dimensional manifolds have zero Euler characteristic. In a previous paper, the author defined a different approach to curvature for arbitrary simplicial complexes, based upon a direct generalization of the angle defect. The generalized angle defect is not zero at the simplices of every odd-dimensional manifold. In this paper we use a sequence based upon the Bernoulli numbers to define a variant of the angle defect for finite simplicial complexes that still satisfies a Gauss-Bonnet-type theorem, but is also zero at any simplex of an odd-dimensional simplicial complex K (of dimension at least 3), such that χ(link(ηⁱ, K)) = 2 for all i-simplices ηⁱ of K, where i is an even integer such that 0 ≤ i ≤ n – 1. As a corollary, an elementary proof is given that any such simplicial complex has Euler characteristic zero.     

A Hochschild Homology Euler Characteristic for Circle Actions

We define an S¹-Euler characteristic, _S ¹(X), of a circle action on a compact manifold or finite complex X. It lies in the first Hochschild homology group HH ₁(G) where G is the fundamental group of X. This _S ¹(X) is analogous in many ways to the ordinary Euler characteristic. One application is an intuitively satisfying formula for the Euler class (integer coefficients) of the normal bundle to a smooth circle action without fixed points on a manifold. In the special case of a three-dimensional Seifert fibered space, this formula is particularly effective.     

On the Euler characteristic of finite unions of convex sets

The Euler characteristic plays an important role in many subjects of discrete and continuous mathematics. For noncompact spaces, its homological definition, being a homotopy invariant, seems not as important as its role for compact spaces. However, its combinatorial definition, as a finitely additive measure, proves to be more applicable in the study of singular spaces such as semialgebraic sets, finitely subanalytic sets, etc. We introduce an interesting integral by means of which the combinatorial Euler characteristic can be defined without the necessity of decomposition and extension as in the traditional treatment for polyhedra and finite unions of compact convex sets. Since finite unions of closed convex sets cannot be obtained by cutting convex sets as in the polyhedral case, a separate treatment of the Euler characteristic for functions generated by indicator functions of closed convex sets and relatively open convex sets is necessary, and this forms the content of this paper.     

Forms of certain hopf algebras

Let Ψ be a field, G a finite group of automorphisms of Ψ, and Φ the fixed field of G. Let H be a Hopf algebra over Ψ. For g ∈ G we define a Hopf algebra H_g which has the same underlying vector space as H and modified operations and show that the tensor product (over Ψ) ?_{g ∈ G} H_g has a Φ-form. As a consequence we see that if n>0 is an integer and Φ is a field of characteristic zero or p>0 with (n,p)=1, then there is a finite dimensional Hopf algebra over Φ with antipode of order 2n.     

Generic muchnik reducibility and presentations of fields

We prove that the subgroup zeta function and the normal zeta function of a finitely generated virtually nilpotent group are finite sums of Euler products of cone integrals over Q and we deduce from this that they have rational abscissa of convergence and some meromorphic continuation. We also define Mal’cev completions of a finitely generated virtually nilpotent group and we prove that the subgroup growth and the normal subgroup growth of the latter are invariants of its Q-Mal’cev completion.     

Galois Module Structure and the γ-Filtration

We describe a general method for calculating equivariant Euler characteristics. The method exploits the fact that the -filtration on the Grothendieck group of vector bundles on a Noetherian quasi-projective scheme has finite length; it allows us to capture torsion information which is usually ignored by equivariant Riemann–Roch theorems. As applications, we study the G-module structure of the coherent cohomology of schemes with a free action by a finite group G and, under certain assumptions, we give an explicit formula for the equivariant Euler characteristic in the Grothendieck group of finitely generated Z[G]-modules, when X is a curve over Z and G has prime order.     

On the Group of p-endotrivial kG-modules

In this paper,we define a group Tp(G) of p-endotrivial kG-modules and a generalized Dade group Dp(G) for a finite group G.We prove that Tp(G) ≌ Tp(H) whenever the subgroup H contains a normalizer of a Sylow p-subgroup of G,in this case,K(G) ≌ K(H).We also prove that the group Dp(G) can be embedded into Tp(G) as a subgroup.