Equivariant Euler characteristics of discriminants of reflection groups

Subgroup posets, Bredon cohomology and equivariant Euler characteristics

For a discrete group $Γ$ satisfying some finiteness conditions we give a Bredon projective resolution of the trivial module in terms of projective covers of the chain complex associated to certain posets of subgroups. We use this to give new results on dimensions of $Eγ$ and to reprove that for virtually solvable groups, $\underline{\cd}Γ=\vcdΓ$. We also deduce a formula to compute the Euler class of $Eγ$ for $Γ$ virtually solvable of type $\FP_\infty$ and use it to compute orbifold Euler characteristics.

Euler characteristics of centralizer subcategories

Let p be a prime number, G a finite group, and A a finite group acting on G. The Brown poset of nonidentity p-subgroups of G is then an A-poset. We investigate the equivariant subposet and the equivariant Euler characteristics and establish a global relation between locally defined Euler characteristics and the number of p-elements of G centralized by A. It is a consequence of this relation that the equivariant version of Brown's theorem holds: The reduced Euler characteristic of the A-equivariant Brown poset is divisible by the p-part of the order of the centralizer of A. The second equivaria…

Generalized Orbifold Euler Characteristic of Symmetric Products and Equivariant Morava K-Theory

We introduce the notion of generalized orbifold Euler characteristic associated to an arbitrary group, and study its properties. We then calculate generating functions of higher order (p-primary) orbifold Euler characteristic of symmetric products of a G-manifold M. As a corollary, we obtain a formula for the number of conjugacy classes of d-tuples of mutually commuting elements (of order powers of p) in the wreath product G wreath S_n in terms of corresponding numbers of G. As a topological application, we present generating functions of Euler characteristic of equivariant Morava K-theories o…

On Equivariant Poincaré Duality, Gysin Morphisms and Euler Classes

The aim of these notes, originally intended as an appendix to a book on the foundations of equivariant cohomology, is to set up the formalism of the $G$-equivariant Poincaré duality for oriented $G$-manifolds, for any connected compact Lie group $G$, following the work of J.-L. Brylinski leading to the spectral sequence $$\mathop{\rm Extgr}\nolimits_{H_G}(H_{G,\rm c} (M),H_G)\Rightarrow H_{G}(M)[d_{M}]\,.$$ The equivariant Gysin functor $(\_)_!:=Ω_{G}(\_)\in\mathcal D^{+}(\mathord{\rm DGM}(H_{G}))$ (resp. $(\_)_{*}:=Ω_{G,\rm c}(\_)$) is then defined in the category of oriented $G$-manifolds an…

图的模空间${\mathbb S}_n$-等变欧拉示性数

The ${\mathbb S}_n$-equivariant Euler characteristic of the moduli space of graphs

本文证明了图的模空间${\mathbb S}_n$-等变欧拉示性数公式，并证明其有理${\mathbb S}_n$-不变上同调在$n$充分大时稳定。

关于$r$个对称群直积等变多项式系统的结式

Resultant of an equivariant polynomial system with respect to a direct product of $r$ symmetric groups

本文针对对称群直积等变多项式系统，建立了结式分解公式，并证明不变判别式可分解为若干较小结式的乘积，从而简化计算。