Equivariant weight filtration for real algebraic varieties with action

Abstract

We show the existence of a weight filtration on the equivariant homology of real algebraic varieties equipped with a finite group action, by applying group homology to the weight complex of McCrory and Parusiński. If the group is of even order, we can not extract additive invariants directly from the induced spectral sequence.Nevertheless, we construct finite additive invariants in terms of bounded long exact sequences, recovering Fichou's equivariant virtual Betti numbers in some cases. In the case of the two-elements group, we recover these additive invariants by using globally invariant chains and the equivariant version of Guillén and Navarro Aznar's extension criterion.

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

“…We compute the equivariant homology of the real 2-dimensional unit sphere X := S 2 in R 3 equipped with two different kind actions of G := Z/2Z, using the spectral sequence E 2 p,q = H p (G, H q (X)) ⇒ H p+q (X; G), induced by the double complex F * ⊗ G C * (X), if F * is a projective resolution of Z 2 over Z 2 [G] (see [5] Chap. VII, see also [18] section 3).…”

“…• The equivariant homology with closed supports H * (X; G) of X is different from the one considered in [18] and [21]. When X is compact, the equivariant cohomology with closed supports H * (X; G) of X coincides with the equivariant cohomology considered in [21] : see remark 4.10 below.…”

“…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).…”

Products of real equivariant weight filtrations

“…We are going to show that the functor Λ • C * G : Sch G c (R) → Ho C + is unique up to filtered quasi-isomorphism (of C + ) in a way similar to theorem 2.3 and proposition 2.6 (see also the homological counterpart Theorem 3.16 of [14]).…”

“…We refer to [1] and [2] for background about group cohomology with coefficients in a module (see also the first part of section 3.1 of [14]). In the following definition, we consider a functor L * that we use to define the cohomology of the group G in a bounded cochain G-complex ; if X is a real algebraic G-variety, the equivariant cohomology of X will be for us the cohomology of the complex L * (C * (X)).…”