Kronecker pairing for a cellular circle generator

Both endpoints of the oriented one-cell attach to the same vertex, so its cellular boundary is zero. There are no two-cells. Thus the degree-one cellular homology is the infinite cyclic group on this cell; its image $z$ under the isomorphism in [F1] is a singular homology generator. The degree-zero group is likewise $\mathbb{Z}$. In particular the cellular cell symbol has only been used to specify a homology class, not as a singular cochain.

The length-zero resolution of $H_0=\mathbb{Z}$ consisting of $\mathbb{Z}$ augmented by identity has zero degree-one Hom group, so ${\mathrm{Ext}}^1(H_0,\mathbb{Z})=0$ by [F2]. The degree-one UCT therefore makes evaluation $\beta :H^1(S^1;\mathbb{Z})\to \mathrm{Hom}(\mathbb{Z}z,\mathbb{Z})$ an isomorphism. The homomorphism $u(sz)=s$ is well-defined because every element has a unique such expression. Define $\alpha ={\beta }^{-1}u$. Then $⟨\alpha ,z⟩=u(z)=1$, and injectivity of $\beta$ makes this class unique.

A singular cocycle representing this class can be obtained exactly as in [F2]: for integral singular cycles $Z_1$ choose the supplied projection ${\pi }_1:C_1\to Z_1$, let $q_1:Z_1\to H_1$ be the quotient, and set $\phi =u{q}_1{\pi }_1$. On a two-boundary, ${\pi }_1$ acts as identity and $q_1$ vanishes, so $\delta \phi =0$. On any singular cycle representing $sz$, its value is $s$. Thus this actual singular cocycle has the required class and evaluation. The construction does not identify a cellular cochain with a singular cochain.

Biadditivity from [F3] yields $⟨r\alpha ,sz⟩=rs$, including zero, negative integers and $r=s=1$. Reversing the cell orientation replaces $z$ by $-z$ and its uniquely normalized dual by $-\alpha$, leaving the normalized value one. The nonempty circle and degree one are fixed; no assertion about a zero-dimensional or empty sphere is involved. AC is inherited from the UCT cycle projection in step 3.1; specifying the single oriented cell adds no infinite choice.