Enumerating prime-power homotopy \(k\)-type (Q1307011)

A homotopy \(k\)-type means the class of all topological spaces that are homotopy equivalent to some given connected polyhedron \(X\) whose \(\pi_i(X)\) are trivial for \(i>k\). For such a \(k\)-type, the author defines its order to be the product \(\prod_{i=1}^k| \pi_i(X)|\) of orders of homotopy groups. He denotes by \(\Lambda (k,m)\) the number of homotopy \(k\)-types of order \(m\). When \(p\) is a prime number, the estimate of \(\Lambda (k,p^n)\) is already known for \(k=1\). In this paper, the author obtains it for \(k\geq 2\). To prove it, he uses the spectral sequences of a fibration and a bisimplicial set.