A solution to de Groot’s absolute cone conjecture

Abstract

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 $n\le 3$ and false for $n\ge 5$. For $n=4$, the absolute cone conjecture is true if and only if the 3-dimensional Poincaré Conjecture is true.