Complex splitting principle with integral injective pullback

Splitting. By [F1] the flag bundle is the iterated projective-bundle tower of $E$ and its metric complements, and the tautological lines satisfy $q^{\ast }E=L_1\oplus \cdots \oplus L_n$. At each stage [F3] applies to the compact complex-projective fiber and the preceding paracompact Hausdorff CGWH CW-type base, so the next base again has all four properties required by [F2].

Each stage has injective structure map. Consider one stage $B_i\to B_{i-1}$ of the tower, the projective bundle of a numerable complex bundle of rank $m\ge 1$ over a CW-type base. By [F2] the cohomology $H^{\ast }(B_i;R)$ is free over $H^{\ast }(B_{i-1};R)$ on the basis $1,x,\dots ,x^{m-1}$; the structure homomorphism $H^{\ast }(B_{i-1};R)\to H^{\ast }(B_i;R)$ is the map $a\mapsto p^{\ast }a$, whose basis coordinates are $(a,0,\dots ,0)$, so it is injective.

Splitting over the flag bundle is step 1.1, so the first two assertions hold.

Injectivity of $q^{\ast }$. The map $q^{\ast }:H^{\ast }(B;R)\to H^{\ast }(\mathrm{Fl}(E);R)$ is the composite of the injective structure maps of the finitely many stages of step 2.1, hence injective.

Finitely many bundles. Ignore the rank-zero bundles, whose pullbacks are already empty sums of lines. For each remaining bundle construct its flag tower over the original CW base $B$, where [F1] applies, and choose its metric once there. Now suppose $F_{j-1}\to B$ has been built, starting with $F_0=B$. Pull the entire original flag tower for $E^{(j)}$ back over $F_{j-1}$, and let $F_j$ be its top. Pullback of a projective stage is the projective bundle of the pulled-back vector bundle: the local identification is $(b,[v])\mapsto [b,v]$ and respects the linear transition maps. Its numeration pulls back with the original chart cover. Thus every stage is licensed by the CW-type extension [F2], without applying the CW-base definition [F1] anew on $F_{j-1}$. Inductively [F3] makes every base paracompact Hausdorff, CGWH, and of CW type. Each such stage has injective cohomology pullback by the module-basis argument of step 2.1. The original splitting is pulled back as an actual bundle isomorphism, so $E^{(j)}$ splits over $F_j$; earlier splittings persist under further pullback. Set $F$ to the final stage. Finite composition gives the required injection simultaneously for all the stated coefficients. If every rank is zero or the family is empty, set $F=B$ and use the identity map.

Boundary cases. For $n=1$ the tower has no projectivization and $q$ is the identity of $B$, so $q^{\ast }$ is injective and $q^{\ast }E=L_1$ with $L_1=E$; the basis $1$ is the trivial basis. For the empty family $m=0$ the assertion is vacuous with $F=B$. The coefficient rings $\mathbb{Z}$ and ${\mathbb{F}}_p$ are nonzero, and the main bundle has positive rank; if an empty base is allowed, all cohomology groups are zero and the injectivity assertion is immediate. No choice beyond the inherited numerability data of [A1] is used, and only finitely many metrics on the original bundles are used, then pulled back through their towers.