The fundamental-group hypothesis in telescope group completion

Remark 5.5.3. The abelian fundamental-group hypothesis is not merely cosmetic. Nikolaus explains that for the commutative monoid \(\bigsqcup _{n\geq 0}\bB \Sigma _n\), the telescope gives \(\Z \times \bB \Sigma _{\infty }\), while the group completion is obtained from this by plus construction. The full Proposition 6 in Nikolaus (2017) gives equivalent criteria in terms of a cyclic-permutation obstruction, and also a useful hypoabelian variant. We will only need the abelian case above.