An elementary approach to local combinatorial formulae for the Euler class of a PL spherical fiber bundle

等变环面几何与Euler-Maclaurin公式

Equivariant toric geometry and Euler-Maclaurin formulae

本文研究了拟射影环面簇的等变motivic Chern类和Hirzebruch特征类，通过等变K-理论和局部化方法推广了多个经典公式，并基于motivic特征类方法给出了全维格多面体的Euler-Maclaurin型公式的代数几何证明及其加权与局部推广。

Euler characteristics of algebraic varieties

This note studies the behavior of Euler characteristics and of intersection homology Euler characterstics under proper morphisms of algebraic (or analytic) varieties. The methods also yield, for algebraic (or analytic) varieties, formulae comparing these two kinds of Euler characteristics. The main results are direct consequences of the calculus of constructible functions and Grothendieck groups of constructible sheaves. Similar formulae for Hodge theoretic invariants of algebraic varieties under morphisms were announced by the first and third authors in \cite{CS1, S}.

用柯西方法证明欧拉公式的一个初等证明

An elementary proof of Euler formula using Cauchy's method

本文用柯西方法初等证明欧拉公式，并推广至环面、射影平面、克莱因瓶等曲面。

A característica de Euler-Poincaré

We introduce the Euler-Poincaré's characteristic with an elementary way and historically. We explain also why one should call Descartes-Poincaré characteristic instead of the Euler-Poincaré's characteristic. All the considered spaces are compact and without boundary. We work essentially on smooth and oriented surfaces . However, we work also on singular and non-oriented surfaces. We provide also the results which may be extended for the case of superior dimension.

Classical formulae on projective surfaces and $3$-folds with ordinary singularities, revisited

As an application of universal polynomials for local and multi-singularities of maps, we revisit classical enumerative formulae of Salmon-Cayley-Zeuthen for projective surfaces and analogous formulae of Segre-(B.)Severi-Roth for projective $3$-folds. In particular, several examples of actual computation are given using universal polynomials for computing weighted Euler characteristics of singularity loci.

Height h detection and connective real k-theory of elementary abelian 2-groups

In this paper, we determine the connective K-cohomology with reality of elementary abelian $2$-groups as a module over $\mathbb{Z}[v_1,a]$, where $v_1$ is the equivariant Bott class and $a$ the Euler class of the sign representation. This gives in particular a new approach to the computation of the connective real K-theory of such groups. The originality here is to make all computations in the $\mathbb{Z}/2$-equivariant stable category, considering only $\mathbb{Z}/2$-equivariant cohomology theories, and to use relative homological algebra over certain subalgebras of the equivariant Steenrod a…