Abstract. We calculate the fibre integrals of the affine hypersurface in a torus in the form of their Mellin transforms. Especially, our method works efficiently for an affine hypersurface defined by a so called “simpliciable” polynomial. The relations between poles of Mellin transforms of fibre integrals, the mixed Hodge structure of the cohomology of the hypersruface, the hypergeometric diffe...

Given a p-adic representation of the Galois group of a local field, we show that its Galois cohomology can be computed using the associated étale (φ,Γ)-module over the Robba ring; this is a variant of a result of Herr. We then establish analogues, for not necessarily étale (φ,Γ)-modules over the Robba ring, of the Euler-Poincaré characteristic formula and Tate local duality for p-adic represent...

Consider a polynomially bounded, o-minimal structure on the field R of reals. A smooth (i.e. of class C∞) definable function φ : U −→ R on an open set U in R determines two closed subsets W := {u ∈ U : φ(u) ≤ 0} and Z := {u ∈ U : φ(u) = 0}. We shall investigate the links of the sets W and Z at the points u ∈ U , which are well-defined up to a definable homeomorphism. It is proven that the Euler...

Let G be a (2,m, n)-group and let x be the number of distinct primes dividing χ, the Euler characteristic of G. We prove, first, that, apart from a finite number of known exceptions, a nonabelian simple composition factor T of G is a finite group of Lie type with rank n ≤ x. This result is proved using new results connecting the prime graph of T to the integer x. We then study the particular ca...

We associate to a good cell decomposition of a manifold M a quadratic algebra and show that the Koszulity of the algebra implies a restriction on the Euler characteristic of M . For a two-dimensional manifold M the algebra is Koszul if and only if the Euler characteristic of M is two.

We recall the notions of weak and strong Euler characteristics on a first order structure and make explicit the notion of a Grothendieck ring of a structure. We define partially ordered Euler characteristic and Grothendieck ring and give a characterization of structures that have non-trivial partially ordered Grothendieck ring. We give a generalization of counting functions to locally finite st...

The Hausmann-Weinberger invariant of a group G is the minimal Euler characteristic of a closed orientable 4–manifold M with fundamental group G. We compute this invariant for finitely generated free abelian groups and estimate the invariant for all finitely generated abelian groups.

The looseness (G) of a triangulation G on a closed surface F 2 is defined as the minimum number k such that for any surjection c : V (G) → {1, 2, . . . , 3+ k}, there exists a face uvw of G which gets three distinct colors c(u), c(v) and c(w). We define min(G) and max(G) as the minimum and the maximum of (G′) taken over all triangulationsG′ on F 2 isomorphic to G as graphs. We shall show that m...

We prove that each polyhedral triangular face free map G on a compact 2-dimensional manifold M with Euler characteristic (M) contains a k-path, i.e. a path on k vertices, such that each vertex of this path has, in G, degree at most 5 2 k if M is a sphere S 0 and at most k 2 5+ p 49?24(M 2 if M 6 = S 0 or does not contain any k-path. We show that for even k this bound is best possible. Moreover,...

An Euler characteristic χ is definable if for every definable function f : X → Y and every r ∈ R, the set {y ∈ Y : χ(f−1(y)) = r} is definable. If R = Z/nZ and M is a finite structure, there is a (unique) Euler characteristic χ : Def(M) → Z/nZ assigning every set its size mod n. This χ is always strong and ∅definable. If M is an ultraproduct of finite structures, then there is a canonical stron...