A solution to de Groot’s absolute cone conjecture

A compactum X is an ‘absolute cone’ if, for each of its points x , the space X is homeomorphic to a cone with x corresponding to the cone point. In 1971, J. de Groot conjectured that each n -dimensional absolute cone is an n -cell. In this paper, we give a complete solution to that conjecture. In particular, we show that the conjecture is true for n ≤ 3 and false for n ≥ 5 . For n = 4 , the absolute cone conjecture is true if and only if the 3-dimensional Poincaré Conjecture is true.