The tautological line over RP∞ has no finite-rank complement

Suppose for contradiction that a rank-$r$ bundle $\eta$ satisfies $\gamma \oplus \eta \cong \mathbb{R}{P}^{\mathrm{\infty }}\times {\mathbb{R}}^{r+1}$. The inclusion of the first summand followed by this isomorphism is a fiberwise-linear embedding $e:\gamma \hookrightarrow \mathbb{R}{P}^{\mathrm{\infty }}\times {\mathbb{R}}^{r+1}$.

The base is in the scope of [F2]. It is paracompact by [F2] and a Hausdorff CW complex by [F3] and [F6]. To see compact generation directly, if $A$ pulls back to a closed set under every compact-Hausdorff test map, then its pullback under every characteristic disk is closed; since a characteristic disk surjects onto its closed cell and the latter is Hausdorff, $A$ meets every closed cell in a closed set, so the weak topology makes $A$ closed. Hausdorffness makes compact images closed, hence the space is also weak Hausdorff. Thus it is CGWH.

The rank-one Schubert symbols in [F3] give exactly one cell in every nonnegative dimension, with $\mathbb{R}{P}^m$ as the $m$-skeleton. By [F4], the differential between any two such cells is zero over ${\mathbb{F}}_2$, since it already occurs in a sufficiently large finite skeleton. Hence [F5] gives $H_k(\mathbb{R}{P}^{\mathrm{\infty }};{\mathbb{F}}_2)\cong {\mathbb{F}}_2$ for every $k\ge 0$, while $H_k(\mathbb{R}{P}^r;{\mathbb{F}}_2)=0$ for $k>r$.

Send $x$ to the image line $e({\gamma }_x)\subseteq {\mathbb{R}}^{r+1}$. In a local nonzero frame $s$ for $\gamma$, this line is represented by the continuous nonzero vector $e(s(x))$, so the resulting map $f:\mathbb{R}{P}^{\mathrm{\infty }}\to {\mathrm{Gr}}_1({\mathbb{R}}^{r+1})=\mathbb{R}{P}^r$ is continuous and $f^{\ast }{\gamma }_r\cong \gamma$. If $j:\mathbb{R}{P}^r\hookrightarrow \mathbb{R}{P}^{\mathrm{\infty }}$ is the standard inclusion, then $(jf{)}^{\ast }\gamma \cong \gamma$.

The identity also pulls $\gamma$ back to itself by [F1]. The injective direction of the classification bijection [F2], applied using step 1.2, therefore gives $jf\simeq {\mathrm{id}}_{\mathbb{R}{P}^{\mathrm{\infty }}}$. This is the sole use of AC in the counterexample.

Put $k=r+1$. On $H_k(-;{\mathbb{F}}_2)$, the map $(jf{)}_{\ast }=j_{\ast }{f}_{\ast }$ is zero because it factors through the zero group $H_k(\mathbb{R}{P}^r;{\mathbb{F}}_2)$ from step 1.3. But [F5] and the homotopy in step 3.1 say that $(jf{)}_{\ast }$ is the identity on the nonzero group $H_k(\mathbb{R}{P}^{\mathrm{\infty }};{\mathbb{F}}_2)\cong {\mathbb{F}}_2$, a contradiction.

Therefore the assumed finite-rank complement $\eta$ cannot exist. The witness is paracompact Hausdorff and CGWH by step 1.2, so it specifically shows that those hypotheses do not replace compactness in the finite-rank complement theorem.