A generalization of the non-triviality theorem of Serre

Abstract:

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 $k$-invariants associated to finite perfect groups. Another application concerns spaces where the cohomology grows like a polynomial algebra on generators in dimension $n$, $2n$, $3n, \ldots$ for a fixed number $n$. We also consider spectra where we prove a non-triviality result in the case of fast growing cohomology groups.