We define here the o-minimal cohomology theory with definably compact supports and prove that this theory is invariant in elementary extensions, in o-minimal expansions and coincides with its topological analogue for o-minimal structures in the real numbers. As an application we show that on definably locally compact definable sets the o-minimal Euler characteristic coincides with the Euler-Poi...

We obtain a precise relation between the Chern-Schwartz-MacPherson class of a subvariety of projective space and the Euler characteristics of its general linear sections. In the case of a hypersurface, this leads to simple proofs of formulas of Dimca-Papadima and Huh for the degrees of the polar map of a homogeneous polynomial, extending these formula to any algebraically closed field of charac...

The Euler characteristic, thought of as a function that assigns a numerical value to every finite simplicial complex, is locally determined in both a combinatorial sense and a geometric sense. In this note we show that not every function that assigns a numerical value to every finite simplicial complex via a linear combination of the numbers of simplices in each dimension is locally determined ...

The topology of the matching complex for the 2×n grid graph is mysterious. We describe a discrete Morse matching for a family of independence complexes Ind(∆n ) that include these matching complexes. Using this matching, we determine the dimensions of the chain spaces for the resulting Morse complexes and derive bounds on the location of non-trivial homology groups for certain Ind(∆n ). Further...

We compute (algebraically) the Euler characteristic of a complex of sheaves with constructible cohomology. A stratified Poincaré-Hopf formula is then a consequence of the smooth Poincaré-Hopf theorem and of additivity of the Euler-Poincaré characteristic with compact supports, once we have a suitable definition of index. AMS classification: 55N33 57R25

We prove an algebraic formula for the Euler characteristic of the Milnor fibres of functions with critical locus a smooth curve on a space which is a weighted homogeneous complete intersection with isolated singularity.

Schanuel has pointed out that there are mathematically interesting categories whose relationship to the ring of integers is analogous to the relationship between the category of finite sets and the semi-ring of non-negative integers. Such categories are inherently geometrical or topological, in that the mapping to the ring of integers is a variant of Euler characteristic. In these notes, I sket...

Let E” be n-dimensional Euclidean space. A molecular space is a family of unit cubes in E”. Any molecular space can be represented by its intersection graph. Conversely, it is known that any graph G can be represented by molecular space M(G) in E” for some n. Suppose that S, and S, are topologically equivalent surfaces in E” and molecular spaces M, and M, are the two families of unit cubes inte...

We study some distributive lattices arising in the combinatorics of lattice paths. In particular, for the Dyck, Motzkin and Schröder lattices we describe the spectrum and we determine explicitly the Euler characteristic in terms of natural parameters of lattice paths. AMS Classification: Primary: 06D05; Secondary: 06A07.

Unlike the well known classical bounds due to Oleinik and Petrovskii, Thom and Milnor on the Betti numbers of (possibly non-symmetric) real algebraic varieties and semialgebraic sets, the above bound is polynomial in k when the degrees of the defining polynomials are bounded by a constant. Moreover, our bounds are asymptotically tight. As an application we improve the best known bound on the Be...