Transversality for Z2-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 ! N transverse to the xed set N Z 2 is given by an element of Sect(BL(M ; N) # M Z 2), i.e. a generalized cohomology class with twisted coeecients. Here BL(M ; 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 Z 2 depends on the is given of a map in the general case f : M ! (N;Z) for which f can be equivariantly homotoped to a transverse map, but only by a large homotopy.