In the paper under review the authors define and study a notion of infinite commutativity for fundamental groups. In particular, they define a space \(X\) to be transfinitely \(\pi_1\)-commutative at a point \(x\in X\) if the infinite permutation of the factors of transfinite loop concatenations based at \(x\) is a homotopy-invariant action.
The sections of this paper are structured as follows. In Section 2, the notation and a review of the relevant theory of the Hawaiian earring group is given. In Section 3, the property of a space \(X\) being transfinitely \(\pi_1\)-commutative at a point \(x \in X\) is stated. The main result of this section characterizes this property in terms of canonical factorizations through the Specker group. In Section 4, several natural examples and situations where the transfinitely \(\pi_1\)-commutative property holds are given. In Section 5, the infinitary commutator subgroup and infinitary abelianization of \(\pi_1(X, x)\) at a subset \(A\subset X\) are defined; and in the case \(A=X\) the authors compare this group to the alternative functorial constructions in the literature.