2021/09/19 by Mandal, Manas, Sankaran, Parameswaran · 1 citation
#Algebraic Topology (math.AT) #FOS: Mathematics
paper · doi:10.48550/arxiv.2109.09107
Let (X,J) be an almost complex manifold with a (smooth) involution σ:X→ X such that fix(σ) is non-empty. Assume that σ is a complex conjugation, i.e, the differential of σ anti-commutes with J. The space P(m,X):=\mathbbSm× X/ ∼ where (v,x)∼ (-v,σ(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 \mathbb Z2-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 σ-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.