Thom isomorphism for the tautological complex line over CP infinity

By definition, ${\mathbb{C}\mathbb{P}}^{\mathrm{\infty }}$ is the weak colimit of complex lines in ${\mathbb{C}}^{N+1}$, so [F1] identifies it with ${\mathrm{Gr}}_1({\mathbb{C}}^{\mathrm{\infty }})$ and $\lambda$ with the pairs $(\mathrm{\ell },v)$, $v\in \mathrm{\ell }$. Its unit vectors therefore form the standard principal $S^1$-bundle. A principal chart with local unit section $s_i(\mathrm{\ell })$ gives the linear chart $(\mathrm{\ell },c{s}_i(\mathrm{\ell }))\mapsto (\mathrm{\ell },c)$ of $\lambda$; hence the support-subordinate numeration in [F2] is also a numeration of $\lambda$. The base is the stated weak CW complex.

A nonzero vector $v$ in a complex fiber orders its underlying real plane by $(v,iv)$. Replacing $v$ by $zv$ for $z\in {\mathbb{C}}^{\times }$ changes this ordered basis by the real matrix of multiplication by $z$, whose determinant is $\mathrm{\mid }z{\mathrm{\mid }}^2>0$. Thus all complex-linear transition functions preserve the corresponding generator in [F3], and these generators define the complex orientation of the underlying real rank-two bundle.

Apply [F4] with $R=\mathbb{Z}$, $n=2$, the numeration from step 1.1, and the orientation from step 1.2. It gives the unique normalized $u_{\lambda }$ and exactly the displayed isomorphism for every $k$; no Euler or characteristic-class identification is used.

The base is nonempty and the coefficients are the nonzero ring $\mathbb{Z}$, so empty-base and zero-ring cases are outside this example. The single complex line, its zero vector and a coordinate-line point ${\mathbb{C}\mathbb{P}}^0\subset {\mathbb{C}\mathbb{P}}^{\mathrm{\infty }}$ are included in steps 1.1–1.2. At the input-degree endpoint $k=0$, the unit maps to $u_{\lambda }$; negative-degree and zero inputs map between zero groups or to zero as covered by [F4]. AC is used only through [A1] in [F4]; the explicit orientation and the supplied numeration require no additional choice.