Related papers: On the Euler characteristic of $S$-arithmetic grou…

We study the $SL(2, \mathbb{C})$ character variety of a Seifert-fibered homology $3$-sphere from the point of view of gauge theory. Namely, we introduce a class of perturbations of the $SL(2,\mathbb{C})$ Chern--Simons functional and prove a…

We show that the covolume of an irreducible lattice in a higher rank semisimple Lie group with the congruence subgroup property is determined by the profinite completion. Without relying on CSP, we additionally show that volume is a…

We use the Euler characteristic of the orbit category of a finite group to establish equivalences between theorems of Frobenius and K.S. Brown and between theorems of Steinberg and L. Solomon.

Let $A$ be a group isomorphic with either $S_4$, the symmetric group on four symbols, or $D_8$, the dihedral group of order 8. Let $V$ be a normal four-subgroup of $A$ and $\alpha$ an involution in $A\setminus V$. Suppose that $A$ acts on a…

We prove an enumerative formula for the algebraic Euler characteristic of Brill-Noether varieties, parametrizing degree d and rank r linear series on a general genus g curve, with ramification profiles specified at up to two general points.…

The Euler characteristic is the only additive topological invariant for spaces of certain sort, in particular, for manifolds with some finiteness properties. A generalization of the notion of a manifold is the notion of a V-manifold. Here…

Let G < SL(V) be a finite group, V is finite dimensional over a field F, p=char F and S(V) is the symmetric algebra of V. We determine when the subring of G-invariants S(V)^G is a polynomial ring. As a consequence, we classify, if F is…

Let K be a complete discretely valued field with residue field k. If char(K) = 0, char(k) = 2 and the 2-rank of k is d, we prove that there exists an integer N depending on d such that the u-invariant of any function field in one variable…

It is well known that the Euler characteristic of the cohomology of a complex algebraic variety coincides with the Euler characteristic of its cohomology with compact support. An old result of G. Laumon asserts that a relative version of…

Let $\Gamma(S)$ be the pure mapping class group of a connected orientable surface $S$ of negative Euler characteristic. For ${\mathscr C}$ a class of finite groups, let $\hat{\pi}_1(S)^{\mathscr C}$ be the pro-${\mathscr C}$ completion of…

We prove that the space of cuspidal quaternionic modular forms on the groups of type $F_4$ and $E_n$ have a purely algebraic characterization. This characterization involves Fourier coefficients and Fourier-Jacobi expansions of the cuspidal…

We use some basic properties of binomial and Stirling numbers to prove that the Euler characteristic is, essentially, the unique numerical topological invariant for compact polyhedra which can be expressed as a linear combination of the…

We consider quotients of the unit cube semigroup algebra by particular $\mathbb{Z}_r\wr S_n$-invariant ideals. Using Gr\"obner basis methods, we show that the resulting graded quotient algebra has a basis where each element is indexed by…

Indices of singular points of a vector field or of a 1-form on a smooth manifold are closely related with the Euler characteristic through the classical Poincar\'e--Hopf theorem. Generalized Euler characteristics (additive topological…

We suggest new realizations of quantum groups corresponding to complex simple Lie algebras, and of affine quantum groups. These new realizations are labeled by Coxeter elements of the corresponding Weyl group and have the following key…

We are interested in stringy invariants of singular projective algebraic varieties satisfying a strict monotonicity with respect to elementary birational modifications in the Mori program. We conjecture that the algebraic stringy Euler…

In this article, we study Euler characteristic techniques in topological data analysis. Pointwise computing the Euler characteristic of a family of simplicial complexes built from data gives rise to the so-called Euler characteristic…

Haken n-manifolds have been defined and studied by B. Foozwell and H. Rubinstein in analogy with the classical Haken manifolds of dimension 3, based upon the the theory of boundary patterns developed by K. Johannson. The Euler…

In this paper we present a construction of the compact form of the exceptional Lie group F4 by exponentiating the corresponding Lie algebra f4. We realize F4 as the automorphisms group of the exceptional Jordan algebra, whose elements are 3…

We define the profinite completion of a C*-algebra, which is a pro-C*-algebra, as well as the pro-C*-algebra of a profinite group. We show that the continuous representations of the pro-C*-algebra of a profinite group correspond to the…