A solution to de Groot’s absolute cone conjecture

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