We study the topology of the Julia set of a quadratic Cremer polynomial P. Our main tool is the following topological result. Let f : U → U be a homeomorphism of a plane domain U and let T ⊂ U be a non-degenerate invariant non-separating continuum. If T contains a topologically repelling fixed point x with an invariant external ray landing at x, then T contains a non-repelling fixed point. Given P, two angles θ, γ are K-equivalent if for some angles x0 = θ, ..., xn = γ the impressions of xi - 1 and xi are non-disjoint, 1 ≤ i ≤ n; a class of K-equivalence is called a K-class. We prove that the following facts are equivalent: (1) there is an impression not containing the Cremer point; (2) there is a degenerate impression; (3) there is a full Lebesgue measure dense Gδ-set of angles each of which is a K-class and has a degenerate impression; (4) there exists a point at which the Julia set is connected im kleinen; (5) not all angles are K-equivalent. © 2006 Elsevier B.V. All rights reserved.