A note on the equivariant cobordism of generalized Dold manifolds

摘要

Let ( X, J) be an almost complex manifold with a (smooth) involution sigma: X - X such that Fix(sigma) not equal empty set. Assume that sigma is a complex conjugation, i.e., the differential of sigma anti-commutes with J. The space P( m, X) := S-m x X/ similar to where ( v, x) similar to (-v, sigma(x)) is known as a generalized Dold manifold. Suppose that a group G congruent to Z(2)(S) acts smoothly on X such that g circle s = sigma circle g for all g is an element of G. Using the action of the diagonal subgroup D= O(1)(m+1) subset of O(m + 1) on the sphere S-m, we obtain an action of G = D x G on S-m x X, which descends to a (smooth) action of G on P(m, X). When the stationary point set X-G for the G action on X is finite, the same also holds for the G action on P(m, X). The main result of this note is that the equivariant cobordism class [P(m, X),G] vanishes if and only if [X, G] vanishes. We illustrate this result in the case when Xis the complex flag manifold, sis the natural complex conjugation and G congruent to (Z(2))(n) is contained in the diagonal subgroup of U(n). (C) 2020 Elsevier B.V. All rights reserved.