zbMATH — the first resource for mathematics

The equivariant cohomology groups of a real algebraic surface and their applications. (English. Russian original) Zbl 0902.55002
Izv. Math. 60, No. 6, 1193-1217 (1996); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 60, No. 6, 101-126 (1996).
Given a non-singular real algebraic variety \(X\) with \(G\) the group generated by the involution induced by complex conjugation the author considers \(G\)-sheaves \({\mathcal A}\) and is interested in the equivariant cohomology groups \(H^{*}(X({\mathbb C}); G, {\mathcal A})\). He extends a five term exact sequence obtained from the first spectral sequence \(I^{p,q}({\mathcal A})\) [cf. A. Grothendieck, Tôhoku Math. J., II. Ser. 9, No. 2, 119-221 (1957; Zbl 0118.26104), p. 201] to an infinite sequence that involves the equivariant cohomology groups and the cohomology of \(X/G\) and of \(X({\mathbb R})\). The sequence is exact if the first spectral sequence converges and the homomorphisms in the sequence are related to the differentials in the \(E_{2}\)-term of \(I^{p,q}({\mathcal A})\). The author gives a precise description of the homomorphisms in case \(X\) is a surface and \({\mathcal A}\) is one of the \(G\)-sheaves \({\mathbb Z/2}, {\mathbb Z}\) or \(\mathbb Z_{-}\), where \({\mathbb Z/2}\) and \({\mathbb Z}\) are the constant sheaves with trivial \(G\)-action, and \({\mathbb Z_{-}}\) is the sheaf that equals \({\mathbb Z}\) but carries the non-trivial \(G\)-action. The author uses his results to give an easy criterion for the convergence of the second spectral sequence \(II^{p,q}({\mathbb Z/2})\) in case \(X\) is a surface with \(X({\mathbb R)} \neq 0\), and he provides an example where \(II^{p,q}({\mathbb Z/2})\) converges but \(II^{p,q}({\mathbb Z})\) does not. Furthermore, using the (exact) sequence for the sheaf \({\mathbb Z_{-}}\), he can find a lower bound for the dimension of \(_{2}Br(X)\), the subgroup of the Brauer group \(Br(X)\) consisting of the elements of order less than or equal to \(2\).

55N91 Equivariant homology and cohomology in algebraic topology
55T99 Spectral sequences in algebraic topology
Full Text: DOI