Constructing Strange Real Algebraic Sets

Recall that algebraic sets are characterized by resolution tower structures [AK1], [AK2]. A topological space with such a structure is called a T R space for short. In low dimensions existence of resolution tower structures can often be detected by combinatorial invariants of the underlying topological spaces. For example, let X be a locally cone-like Euler stratified space (i.e., a locally conelike stratified set so that the Euler characteristics of the links of all strata are even). Then for every component of the codimension one stratum U of X we define ~ x ( U ) to be the number of points in the link of U. For every component of the codimension two stratum W of X we define a i ( W ) to be the number of vertices v of the link L of W so that ~L(V) = i rood 8, where i = 0 , . . . , 7. Then define