Sausages and Butcher Paper

Abstract

AbstractFor each \(d>1\) the shift locus of degree d, denoted \({\mathcal S}_d\), 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 \({\mathcal S}_d\) as a contractible complex of spaces, 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 \({\mathcal S}_d\) has the homotopy type of a CW complex of real dimension \(d-1\); and that \({\mathcal S}_3\) and \({\mathcal S}_4\) are \(K(\pi ,1)\)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 \(\mathbb C^2\)) which are homotopic to locally \(\text {CAT}(0)\) complexes; in particular they are \(K(\pi ,1)\)s.