Integral homology of a wedge of higher spheres has its cell basis

For a finite set $J$ and $j\in J$, put $U={\bigvee }_{k\in J\setminus \{j\}}{S}_k^n$, retaining $b$ when this set is empty. This is a nonempty CW subcomplex by [F1]. Collapsing the last sphere gives a continuous retraction $r:W\to U$, since its characteristic-disk restrictions are the identity on the remaining disks and constant on the last. The quotient $W\mathrm{/}U$ is canonically $S_j^n$: its one remaining characteristic disk has its entire boundary collapsed, and the map-out test is precisely that of this sphere. Under this identification the quotient map is $p_j$. Write $k:U\hookrightarrow W$, $s={\iota }_j$, and $j_S:H_i(S_j^n)\to H_i(S_j^n,\{b\})$. For $i>0$, [F2] and [F4] make $t=j_S^{-1}(p_j{)}_{\ast }:H_i(W,U)\to H_i(S_j^n)$ an isomorphism. If $j_W:H_i(W)\to H_i(W,U)$ is the pair map, naturality on singular chains gives $t{j}_W=(p_j{)}_{\ast }$, since both sides postcompose with $p_j$.

For any set $J$, every integral singular chain $c$ in $W$ is supported in a finite subwedge. Indeed [F6] writes $c$ as a finite sum of singular simplices. Their domains are compact by [F6], so each image lies in a finite CW subcomplex. The union of the finitely many resulting cell sets, enlarged by $b$, is a finite subwedge of this particular CW structure. Only finitely many witnesses are needed for this one chain. Inclusions of subspaces induce injective singular chain maps: distinct maps into the subspace remain distinct after its set-theoretic inclusion, so their finite formal sums stay distinct. Consequently a chain supported in a subwedge is a cycle there exactly when it is a cycle in $W$, and a displayed boundary equation between supported chains also holds in the subwedge.

For $i>0$, $k_{\ast }$ is injective because $r_{\ast }{k}_{\ast }=1$. Also $(p_j{)}_{\ast }{s}_{\ast }=1$. Exactness of [F3] and step 1.1 give $\ker (p_j{)}_{\ast }=\mathrm{im}{k}_{\ast }$. For any $z\in H_i(W)$, $z-s_{\ast }(p_j{)}_{\ast }z$ is in that kernel, so is uniquely $k_{\ast }x$. Applying $r_{\ast }$ gives $x=r_{\ast }z$, since $rs:S_j^n\to U$ is constant and is zero on positive homology: it factors through a point, whose positive homology vanishes as used in [F4]. Therefore $z=k_{\ast }{r}_{\ast }z+s_{\ast }(p_j{)}_{\ast }z\mathrm{.}$ Conversely $r_{\ast }{k}_{\ast }=1$, $(p_j{)}_{\ast }{s}_{\ast }=1$, and both cross composites are zero by the same constant-map argument. Thus $(k_{\ast },s_{\ast })$ gives an isomorphism $H_i(U)\oplus H_i(S_j^n)\to H_i(W)$ with inverse $(r_{\ast },(p_j{)}_{\ast })$. This also proves the splitting in degree one without assuming anything about a negative-degree group.

Induction on the finite cardinality of $J$ now proves that the positive homology of a finite wedge is the direct sum of the homology of its sphere summands, with inclusions as the forward map and the collapsing projections as inverse. The initial empty wedge is a point and has zero positive homology. Each induction step is precisely step 2.1. In positive degree $i\mathrm{\ne }n$, all summand groups vanish by [F5]; in degree $n$, each is the copy of $\mathbb{Z}$ specified by its supplied orientation. Hence the formulas in the statement hold for finite $J$, including a singleton. There is no choice of an ordering over all finite subsets: induction proves the unique maps specified by the coordinate formulas.

For arbitrary $J$, every class $z\in H_i(W)$ with $i>0$ has a finite cycle representative in a finite subwedge by step 1.2. If $i\mathrm{\ne }n$, step 3.1 makes it a boundary in that subwedge and hence in $W$. If $i=n$, step 3.1 expresses it as a finite sum of the sphere orientation classes, proving surjectivity of $\mathrm{\Psi }$. This map is well defined by finite sums in the abelian homology group and the finite-support group construction [F1]. If a finite vector $a$ maps to zero, represent its finite sum by orientation cycles in those finitely many spheres. Its image is the boundary of one finite chain in $W$. Step 1.2 puts that chain in a finite subwedge; enlarge it by the finite support of $a$. The chain boundary equation holds already there, so step 3.1 forces every coefficient of $a$ to be zero. This proves injectivity.

A cycle for $z\in H_n(W)$ is supported in a finite subwedge $W_{J_0}$ by step 1.2. For $j\mathrm{\notin }{J}_0$, $p_j$ is constant there, so $(p_j{)}_{\ast }z=0$ in positive degree. For $j\in J_0$, the finite computation of step 3.1 identifies its coefficient with exactly $(p_j{)}_{\ast }z=a_j[S_j^n]$. This proves the asserted inverse and finite support; the coefficients depend only on $z$ because induced homology maps are well defined. Finally $W$ is nonempty and path connected by [F1], including the stipulated empty wedge, so [F5] gives $H_0(W)=\mathbb{Z}$. Thus degree zero is one shared component, not a sum over $J$. Zero vectors and zero cycles were included in the finite argument, and no first-degree exception is hidden: $H_1(W)=0$ since $n\ge 2$. Every arbitrary-index passage used a single finite chain or bounding chain, and orientations were supplied, so no AC was used.