Hurewicz calculation for a wedge of simply connected spheres

By [F2], $W$ has a CW structure with one vertex and one $n$-cell for each $j$, each attached by its constant boundary. It is path connected, and ${\pi }_i(W,b)=0$ for $0<i<n$. Thus $W$ is $(n-1)$-connected and meets [F1]'s CW and nonemptiness hypotheses. In particular $n=2$ gives simple connectivity, rather than assuming it from an unstated wedge principle. By [A1] and [F1], $h:{\pi }_n(W,b)\to H_n(W;\mathbb{Z})$ is an isomorphism.

Formula [F4] gives $h([{\iota }_j])=({\iota }_j{)}_{\ast }[S_j^n]=e_j\mathrm{.}$ Since $h$ is a homomorphism, it follows for every integer vector $a=(a_j{)}_{j\in J}$ that $h({\sum }_{j\in J}{a}_j[{\iota }_j])={\sum }_{j\in J}{a}_j{e}_j\mathrm{.}$ The sum on the left is well defined: $h$ is an injective homomorphism into the abelian homology group, so its source is abelian (the image of each commutator is zero, hence the commutator itself is the identity). Finite sums are therefore independent of the order, and negative coefficients mean inverse classes.

For $\alpha \in {\pi }_n(W,b)$ define $c_j(\alpha )$ by $(p_j{)}_{\ast }h(\alpha )=c_j(\alpha )[S_j^n]\mathrm{.}$ These integers are unique by [F3], and they form a vector in the finite direct sum because $J$ is finite. Let $T(a)={\sum }_j{a}_j[{\iota }_j]$. The two composites are identities: step 2.1 and the coordinate inverse of [F3] give $c(T(a))=a$; conversely [F3] expresses $h(\alpha )={\sum }_j{c}_j(\alpha )e_j=h(T(c(\alpha )))$, and injectivity of $h$ gives $T(c(\alpha ))=\alpha$. Both maps are homomorphisms by [F3], [F4] and finite additivity. This proves the stated isomorphism with precisely the inclusion basis, not just an abstract equality of ranks.

A singleton $J$ recovers one sphere and its identity generator. The zero vector corresponds to the constant class; negative vectors correspond to inverse classes by step 2.1. Empty $J$ is excluded in the stated example, although [F2] and [F3] consistently assign it a point and a zero positive group. The restriction $n\ge 2$ is required for [F1]'s higher Hurewicz isomorphism; no free-abelian claim for a wedge of circles is being made. The AC cost of this derivation is exactly [A1]; choosing an ordering of every finite set or a family of orientation representatives was not used. The claimed connectivity and both inverse formulas are now proved.