A Generalization of the Non-triviality Theorem of Serre

We generalize the classical theorem of Serre on the non-triviality of infinitely many homotopy groups of 1-connected finite CW-complexes to CW-complexes where the cohomology groups either grow too fast or do not grow faster than a certain rate given by connectivity. For example, this result can be applied to iterated suspensions of finite Postnikov systems and certain spaces with finitely generated cohomology ring. In particular, we obtain an independent, short proof of a theorem of R. Levi on the non-triviality of kinvariants associated to finite perfect groups. Another application concerns spaces where the cohomology grows like a polynomial algebra on generators in dimension n, 2n, 3n, . . . for a fixed number n. We also consider spectra where we prove a non-triviality result in the case of fast growing cohomology groups. A classical theorem of J.P. Serre (for p = 2, [S]) and Y. Umeda (for odd p, [U]) states that a 1-connected finite CW-complex X with non-trivial cohomology mod p has infinitely many non-trivial homotopy groups mod p. In [MN], C.A. McGibbon and J.A. Neisendorfer proved a stronger result (the existence of p-torsion elements in infinitely many dimensions for finite dimensional spaces) using H. Miller’s theorem on the Sullivan conjecture. Using the same methods from analytical number theory as in the original proof of Serre, we show a straightforward generalization of the Serre non-triviality theorem to certain infinite CW-complexes, which seems not to be in the literature. The point is that the cohomology of a finite Postnikov section has a certain growth rate depending of the highest homotopy group. Thus, spaces where cohomology grows faster or slower cannot have a finite Postnikov system. We need some notation: Definition 1. Let X be a connected CW-complex and p a prime. We call X of finite type if Hk(X ;Z) is finitely generated for all k > 0. Then we can define the mod p Poincaré series as the formal power series with integer coefficients h(X, t) := ∑