Sausages and Butcher Paper

Abstract

For each d > 1 the shift locus of degree d, denoted S d , is the space of normalized degree d polynomials in one complex variable for which every critical point is in the attracting basin of infinity under iteration. It is a complex analytic manifold of complex dimension d − 1.We are able to give an explicit description of S d as a complex of spaces over a contractible Ãd−2 building, and to describe the pieces in two quite different ways:(1) (combinatorial): in terms of dynamical extended laminations; or (2) (algebraic): in terms of certain explicit 'discriminant-like' affine algebraic varieties. From this structure one may deduce numerous facts, including that S d has the homotopy type of a CW complex of real dimension d − 1; and that S 3 and S 4 are K(π, 1)s.The method of proof is rather interesting in its own right. In fact, along the way we discover a new class of complex surfaces (they are complements of certain singular curves in C 2 ) which are homotopic to locally CAT(0) complexes; in particular they are K(π, 1)s.