Equivariant virtual Betti numbers

Abstract

Abstract. We define a generalised Euler characteristic for arc-symmetric sets endowed with a group action. It coincides with the Poincaré series in equivariant homology for compact nonsingular sets, but is different in general. We put emphasis on the particular case of Z/2Z, and give an application to the study of the singularities of Nash function germs via an analog of the motivic zeta function of Denef and Loeser.

Quotients and invariants of $$\mathcal {AS}$$-sets equipped with a finite group action

“…We will suppose furthermore that the projective Zariski closure of X in P d (R) is equipped with an action of G by biregular isomorphisms, and that X is globally stabilized under this action : we will say that X is a G-AS-set (see [9], paragraph 3.1.1). Now, recall that the real algebraic variety P d (R) can be biregularly embedded into a compact algebraic subset of R (d+1) 2 ([3],Theorem 3.4.4), so that X can be supposed to be, up to equivariant biregular isomorphism, an AS-subset of R (d+1) 2 such that its (affine) Zariski closure (in R (d+1) 2 ) is compact and equipped with an action of G by biregular isomorphisms which globally preserves X (X is a boolean combination of compact arc-symmetric sets of R (d+1) 2 : see Remark 3.5 of [11]).…”

“…Another equivariant homology of the sphere was computed in [9] (Example 2.8) and [18] (Example 3.13), and the equivariant cohomology of the circle was computed in [21] (Example 3.5).…”

“…From this spectral sequence E(X; G), we recover invariants with values in Z which are additive with respect to equivariant inclusions of G-ASsets, and coincide with equivariant homology on compact nonsingular G-AS-sets : we call them the equivariant virtual Betti numbers of X. They are different from the equivariant virtual Betti numbers of [9].…”

“…Proof. We use an induction on the dimension of X (see also [9], proof of Proposition 3.10 for instance).…”

Complexe de poids des variétés algébriques réelles avec action

“…Supposons que la G-variété algébrique réelle X soit compacte, et que l'action de G sur X soit libre. Alors le quotient X/G (qui désigne par abus de notation le quotient de l'ensemble des points réels de X par l'action de G restreinte) est un ensemble symétrique par arcs ( [5] Proposition 3.15) et, pour tous k, α ∈ Z, on a un isomorphisme…”