Cohomology of generalized Dold spaces

Let (X, J) be an almost complex manifold with a (smooth) involution sigma : X -> X such that Fix(sigma) not equal circle divide. Assume that sigma is a complex conjugation, i.e., the differential of sigma anti-commutes with J. The space P(m, X) := S(m)xX/similar to where (v, x) similar to(-v, sigma(x)) was referred to as a generalized Dold manifold. The above definition admits an obvious generalization to a much wider class of spaces where X, S are arbitrary topological spaces. The resulting space P(S, X) will be called a generalized Dold space. When S and X are CW complexes satisfying certain natural requirements, we obtain a CW-structure on P(S, X). Under certain further hypotheses, we determine the mod 2 cohomology groups of P(S, X). We determine the Z(2)-cohomology algebra when X is (i) a torus manifold whose torus orbit space is a homology polytope, (ii) a complex flag manifold. One of the main tools is the Stiefel-Whitney class formula for vector bundles over P(S, X) associated to sigma-conjugate complex bundles over X when the S is a paracompact Hausdorff topological space, extending the validity of the formula, obtained earlier by Nath and Sankaran, in the case of generalized Dold manifolds. (c) 2022 Elsevier B.V. All rights reserved.