Euler class of the universal oriented two-plane

Base hypotheses. The real Grassmannian and $C$ are CW complexes by [F1]. The double cover $B\to {\mathrm{Gr}}_2({\mathbb{R}}^{\mathrm{\infty }})$ is numerable by a partition subordinate to evenly covered neighborhoods. Its fiber is the compact two-point CW complex, and the CW base is paracompact Hausdorff and CGWH. The separate paracompactness, compact-generation, and CW-type clauses of [F3] therefore make $B$ paracompact Hausdorff CGWH of CW type; Hausdorff also makes it weak Hausdorff. Thus both $B$ and $C$ satisfy the classification and Thom base hypotheses. Partitions subordinate to the tautological linear charts make their bundles numerable; the same holds for their pullbacks. These are the uses of AC and its consequence DC.

Complex structures on planes. On an oriented Euclidean plane let $J$ be the positive quarter turn. In each positive orthonormal frame it has the same matrix, which commutes with every transition rotation in $\mathrm{SO}(2)$; hence $J$ is continuous and makes ${\gamma }_2^{+}$ a complex line $\eta$. If $T$ is any orientation-preserving real isomorphism between complex lines, its complex-linear part is $T^{1,0}=(T-J_{\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{t}}T{J}_{\mathrm{s}\mathrm{o}\mathrm{u}\mathrm{r}\mathrm{c}\mathrm{e}})\mathrm{/}2$. In complex coordinates $T(z)=az+b\bar{z}$, and ${\det }_{\mathbb{R}}T=\mathrm{\mid }a{\mathrm{\mid }}^2-\mathrm{\mid }b{\mathrm{\mid }}^2>0$ implies $a\mathrm{\ne }0$. Therefore $T^{1,0}$ is a complex-linear isomorphism. This formula is continuous and independent of frames, so it also applies to bundle isomorphisms.

Classifying maps. By [F2], choose $f:C\to B$ classifying ${\gamma }_{\mathbb{R}}$ with its complex orientation, and $g:B\to C$ classifying $\eta$. The complex charts of $\eta$ constructed in step 1.2 admit a subordinate partition by step 1.1. The composite $fg$ classifies ${\eta }_{\mathbb{R}}={\gamma }_2^{+}$, so $fg\simeq 1_B$. The oriented isomorphism $f^{\ast }{\eta }_{\mathbb{R}}\cong {\gamma }_{\mathbb{R}}$ yields a complex isomorphism $f^{\ast }\eta \cong \gamma$ by step 1.2; therefore $gf\simeq 1_C$. Composition of pullbacks is identified fiberwise by $(x,(f(x),v))\mapsto (x,v)$. Thus $f$ is a homotopy equivalence, without asserting a homeomorphism between the chosen Grassmannian models.

Integral generator. By [F7], $f^{\ast }:H^2(B;\mathbb{Z})\to H^2(C;\mathbb{Z})$ is an isomorphism. Naturality [F4] gives $f^{\ast }e({\gamma }_2^{+})=e({\gamma }_{\mathbb{R}})=u$, which generates the target by [F4]. Hence $e({\gamma }_2^{+})$ generates the source.

Sign on ${\mathbb{C}\mathbb{P}}^1$. Put $M={\mathbb{C}\mathbb{P}}^1$, $L={\gamma }^{\ast }{\mathrm{\mid }}_M$, and $p=[1:0]$. The functional $(z_0,z_1)\mapsto z_1$ restricts on each line to a section $s$ of $L$, vanishing only at $p$. In the affine coordinate $w=z_1\mathrm{/}{z}_0$ and the dual tautological frame $e(w)$, this section is $s(w)=we(w)$. Choose a small closed coordinate disk $D=\{\mathrm{\mid }w\mathrm{\mid }\le \epsilon \}$ and write $a(w)=\mathrm{\parallel }e(w)\mathrm{\parallel }>0$. Define a section $\tilde{s}$ of the unit disk bundle by $\tilde{s}(q)=s(q)\mathrm{/}(\epsilon a(q))$ for $q\in D$ and $\tilde{s}(q)=s(q)\mathrm{/}\mathrm{\parallel }s(q)\mathrm{\parallel }$ for $q\mathrm{\notin }\mathrm{int}D$. The two formulas agree on $\mathrm{\partial }D$, where $\mathrm{\parallel }s(w)\mathrm{\parallel }=\epsilon a(w)$, so $\tilde{s}$ is continuous; it is sphere-valued on $A=M\setminus \mathrm{int}D$. Hence it is a genuine map of pairs $(M,A)\to (D(L),S(L))$ and pulls the normalized Thom class $U$ back to $\alpha \in H^2(M,A;\mathbb{Z})$. The absolute image of $\alpha$ is ${\tilde{s}}^{\ast }{j}^{\ast }U=e(L_{\mathbb{R}})=c_1(L)$ by [F5]: as maps to $D(L)$, $\tilde{s}$ and the zero section are joined by the fiberwise straight-line homotopy $(1-t)\tilde{s}$. Excision (equivalently the quotient identification $M\mathrm{/}A\cong D\mathrm{/}\mathrm{\partial }D$) restricts $\alpha$ to the class induced on $(D,\mathrm{\partial }D)$ by $w\mapsto w\mathrm{/}(\epsilon a(w))$ in the frame $e(w)$. The positive scalar factor preserves the complex orientation and the boundary map has degree $+1$, so this is the positive local orientation class. The complex-oriented fundamental class restricts to that generator, and the evaluation pairing gives $⟨c_1(L),[M]⟩=+1$. Thus the positive generator is $x=c_1({\gamma }^{\ast })$, while [F5] gives $u{\mathrm{\mid }}_M=c_1(\gamma )=-x$. Complex orientation of the bundle fiber therefore does not make its Euler number positive on the complex-oriented base.

Reduction and boundary. The admissibility and numerability checked in step 1.1 allow [F6] to give ${\rho }_2(e({\gamma }_2^{+}))=w_2({\gamma }_2^{+})$. Its pullback and then restriction to ${\mathbb{C}\mathbb{P}}^1$ is the nonzero reduction of $-x$ by [F6] and [F7], proving nonvanishing. This is a positive-rank computation on a nonempty base. For comparison, the trivial oriented two-plane over a point has Euler class zero since $H^2(\mathrm{p}\mathrm{t};\mathbb{Z})=0$ (its normalized singular cochain complex has no positive degrees); the rank-zero Euler unit lies instead in degree zero.