COMPLEX MONODROMY AND CHANGING REAL PICTURES

Abstract

The purpose of this note is to generalise, and give a more illuminating proof, of a theorem of [13] (Theorem 1.1 below). Before stating it, we provide some introductory information. Consider the following two sequences of pictures: in each we see a 1-parameter family X_ℝ,t of real algebraic hypersurfaces, which undergoes a bifurcation when the parameter t is equal to 0. Note that in Figure 1, both (i) (a) and (i) (b), and in (ii) (b), the surface X_ℝ,t has a purely 1-dimensional part, which we have indicated with a dotted line, and that in (i) (b) we have drawn a curve vertically along the middle of the surface to make clearer the way it passes through itself. The reader will observe that in (a) the surface X_ℝ,t is homotopically a 2-sphere when t>0 and a 0-sphere when t<0, while in (b) X_ℝ,t is a homotopy 1-sphere both for t<0 and t>0.

Such sequences are typical in singularity theory; each is in fact the family of algebraic closures of images of a versal deformation of a codimension 1 singularity of mapping.

Now suppose that the complexification X_{[Copf ],t} is a homotopy n-sphere. In [13] the second author pointed out that it follows that X_ℝ,t is a homotopy sphere for t≠0 (allowing the empty set as a −1-sphere). Indeed, in the local situation, or globally in the weighted homogeneous case, there are well-defined integers k₊ and k₋ between −1 and n such that X_ℝ,t≃S^k+ for t>0 and X_ℝ,t≃S^k− for t<0.

We describe X_ℝ,t for t∈ℝ−0 as ‘good’ if the homotopy dimension of X_ℝ,t is equal to n. In this case the inclusion X_ℝ,t[rarrhk ]X_t is a homotopy equivalence [13, 1.1].