Summary: For 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 \(K(\pi ,1)\mathrm{s}\). 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 \(\mathrm{CAT}(0)\) complexes; in particular they are \(K(\pi ,1)\mathrm{s}\).
For the entire collection see [Zbl 1496.14003].