Elliptic spaces with the rational homotopy type of spheres

The Milnor-Moore theorem shows that the Q-elliptic spaces are precisely those spaces which have the rational homotopy type of finite, 1-connected CW complexes and have finite total rational homotopy rank. This class of spaces is important because of the dichotomy theorem (the subject of the book [8]) which states that a finite, 1-connected complex either has finite total rational homotopy rank or the rational homotopy groups have exponential growth when regarded as a graded vector space. Moreover, elliptic spaces are the subject of the Moore conjectures, asserting that the homotopy groups of a finite, 1-connected CW complex have finite exponent at all primes if and only if it is Q-elliptic. The p-elliptic spaces form a sub-class of the class of Q-elliptic spaces. A p-elliptic space Z is known to satisfy the following important properties [10, 11].