A generalization of the non-triviality theorem of Serre (Q2781280)

The author considers to generalize the result of \textit{J. P. Serre} (for \(p=2)\) and \textit{Y. Umeda} (for odd \(p)\) on the non-triviality of infinitely many homotopy groups of 1-connected finite CW-complexes to infinite CW-complexes. Let \(X\) be a 1-connected CW-complex of finite type. The main result is then the following: (1) If \(\rho(X) <1\), or \(\rho(X)=1\) and \(\text{lgr}(X)= \infty\), then \(\pi_*(X) \otimes\mathbb{Z}/p\) is non-trivial in arbitrarily high dimensions, and (2) if \(\rho(X)\geq 1\) and \(\text{lgr}(X) <\text{conn} (X)\) this also holds true. More strongly it holds that the Postnikov tower for \(X\) has infinitely many non-vanishing \(k\)-invariants. Furthermore, in the case that \(H^k(X; \mathbb{Z})\) is finite for \(k\geq 2\) this is also true for \(\text{lgr} (X)=\text{conn}(X)\) (theorem 2). Here \(\text{conn}(X)\) denotes the connectivity mod \(p\) of \(X\), \(\rho(X)\) denotes the radius of convergence (of Cauchy type) of the mod \(p\) Poincaré series \(h(X,t)= \Sigma_k h^k(X)t^k\), \(h^k(X)= \dim H^k(X; \mathbb{Z}/p)\), and \(\text{lgr}(X)\) is a function which indicates a certain growth rate of the mod \(p\) cohomology of \(X\) with \(\rho(X) \geq 1\) (we omit the precise definition). The proof is heavily based on the result due to Serre and Umeda such that for a non-trivial finitely generated group \(\pi\), \(\rho(K (\pi,n))=1\) and \(\text{lgr} (K(\pi,n))= n-1\) or \(n\) according as \(\pi\) is free or not free.NEWLINENEWLINENEWLINEIn the latter half of the paper the author considers what spaces the main result above can be applied to (corollaries 1-4). It is shown, for example, that we have the classifying spaces \(BO\), \(BU\), \(BSp\), \(BSpin\), etc. and their iterated suspensions as concrete examples. Finally the author deals with a generalization to connected spectra of finite types (theorem 3 and corollary 5).