The type of a torsion free finite loop space (Q1183658)

The types conjecture asserts that every simply connected loop space of the homotopy type of a compact \(CW\)-complex has the rational cohomology ring of a compact Lie group. Such a cohomology ring is an exterior algebra \(\Lambda(x_ 1,\dots,x_ r)\) with generators \(x_ i\) in odd dimensions \(d_ i\), where \(3\leq d_ 1\leq d_ 2\leq\dots d_ r\). The authors verify the types conjecture for loop spaces whose integral cohomology is free of 2-torsion and whose rational cohomology satisfies one of the two conditions below: (1) Each \((d_ i+1)\) is divisible by 4 and each \(d_ i<60\). (2) Each \((d_ i+1)\) is divisible by 4 and the number of \(x_ i\)'s in dimensions 43 and 51 is greater than or equal to the number of \(x_ i\)'s in dimensions 47 and 55, respectively.