2020/02/20 by Nath, Avijit, Sankaran, Parameswaran
#57R25 #Algebraic Topology (math.AT) #FOS: Mathematics
paper · doi:10.48550/arxiv.2002.08692
Let (X,J) be an almost complex manifold with a (smooth) involution σ:X→ X such that Fix(σ)≠ ∅. 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)) is known as a generalized Dold manifold. Suppose that a group G≅ \mathbb Z2s acts smoothly on X such that g∘ σ=σ∘ g for all g∈ G. Using the action of the diagonal subgroup D=O(1)m+1⊂ O(m+1) on the sphere \mathbb Sm for which there are only finitely many pairs of antipodal points that are stablized by D, we obtain an action of \mathcal G=D× G on \mathbb Sm× X, which descends to a (smooth) action of \mathcal G on P(m,X). When the stationary point set XG for the G action on X is finite, the same also holds for the \mathcal G action on P(m,X). The main result of this note is that the equivariant cobordism class [P(m,X),\mathcal G] vanishes if and only if [X,G] vanishes. We illustrate this result in the case when X is the complex flag manifold, σ is the natural complex conjugation and G≅ (\mathbb Z2)n is contained in the diagonal subgroup of U(n).