Products of real equivariant weight filtrations

Abstract

We first show the existence of a weight filtration on the equivariant cohomology of real algebraic varieties equipped with the action of a finite group, by applying group cohomology to the dual geometric filtration. We then prove the compatibility of the equivariant weight filtrations and spectral sequences with Künneth isomorphism, cup and cap products, from the filtered chain level. We finally induce the usual formulae for the equivariant cup and cap products from their analogs on the non-equivariant weight spectral sequences.

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

“…• 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.…”

“…Consequently, if X is compact, H * (X; G) = H * (G, C * (X)) ( [5] Chap. VII, Proposition 5.2), and by dualization, H * (X; G) = H * (G, C * (X)) (this was the definition of the equivariant cohomology considered in [21], Definition 3.3).…”

“…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).…”

“…Remark 5.22. If X is compact nonsingular, we have a Poincaré duality isomorphism between H * (X; G) and the equivariant homology considered in [9] (see [26] III Theorem 4.2, [9] 2.3.5 and also [21] Remark 4.26) so that β(X; G)(u) = u d β G (X)(u −1 )…”