Homotopy classification of higher homotopy commutative loop spaces with finitely generated cohomology (Q1588319)

Suppose that \(p\) is an odd prime. The author studies a simply connected mod \(p\) loop space \(X\) such that its mod \(p\) cohomology \(H^*(X)\) is finitely generated. He proves that if \(X\) is a \(C_n\)-space in the sense of Williams, then it is the total space of \(C_n\)-fibration over a finite \(C_n\)-space. Using this result he also obtains a necessary and sufficient condition for \(C_2\)-spaces. Moreover, he proves that if \(X\) is a simply connected \(C_p\)-space such that its mod \(p\) cohomology is finitely generated it is homotopy equivalent to the finite product of infinite dimensional complex projective spaces.