Transversality for \(\mathbb{Z}_2\)-orbifolds
Equivariant transversality phenomena are studied for PL locally linear actions of the group with two elements. In this setting with stability assumptions, the obstruction to making a given map \(f:M\to N\) transverse to the fixed set \(N^{\mathbb{Z}_2}\) is given by an element of \(\text{Sect} (BL(\nu_M,\nu_N) \downarrow M^{\mathbb{Z}_2})\) i.e. a generalized cohomology class with twisted coefficients. Here \(BL(\nu_M,\nu_N)\) is a spectrum whose homotopy groups are appropriate generalizations of the Browder-Livesay obstructions to desuspension of homotopy real projective spaces. Which spectrum is used and how it twists over \(M^{\mathbb{Z}_2}\) depends on the \(\mathbb{Z}_2\) block bundles \(\nu_M=\nu (M^{\mathbb{Z}_2} \subset M)\) and \(\nu_N=\nu (N^{\mathbb{Z}_2}\subset N)\). An example is given of a map in the general case \(f:M\to(N,Z)\) for which \(f\) can be equivariantly homotoped to a transverse map, but only by a large homotopy.
scientific article; zbMATH DE number 1336975