Complex flag bundle and Chern roots

Assume AC. Let $E\to B$ be a numerable complex rank-$n$ bundle with $n\ge 1$ over a path-connected paracompact Hausdorff CW complex, equipped with a Hermitian metric $h$, which exists by Numerable vector bundles admit bundle metrics.

A complete flag in a fiber $E_b$ is a chain of complex subspaces $0=V_0\subset V_1\subset V_2\subset \cdots \subset V_n=E_b,{\dim }_{\mathbb{C}}{V}_i=i\mathrm{.}$ The flag bundle $q:\mathrm{Fl}(E)\to B$ is the bundle of complete flags: its points are the pairs $(b,V_{\bullet })$, topologized by the iterated projective-bundle construction, which exhibits it as a fiber bundle with fiber the full flag manifold $U(n)\mathrm{/}{T}^n$. Concretely, put $B_0:=B$ and $E^{(0)}:=E$. For $r=1,\dots ,n-1$, let $B_r:=P(E^{(r-1)})$ with projection ${\pi }_r:B_r\to B_{r-1}$, let $L_r\subseteq {\pi }_r^{\ast }{E}^{(r-1)}$ be its tautological line, and let $E^{(r)}:=L_r^{\perp }$ inside ${\pi }_r^{\ast }{E}^{(r-1)}$. Pulling the earlier $L_j$ through the later projections and taking $L_n:=E^{(n-1)}$ gives the $n$ ordered orthogonal lines. The composite $q:B_{n-1}\to B$ is $\mathrm{Fl}(E)$. All intermediate bases are paracompact Hausdorff CGWH spaces of CW type by Compact-fibre bundle totals preserve paracompactness, and CW type under CW-type hypotheses. At every stage use the CW-type construction established in Integral complex projective bundle theorem, not an assertion that $B_r$ itself is a CW complex. The projective charts are numerated by the partition for $E^{(r-1)}$. At every compact projective-fiber stage, the same lemma supplies the CGWH conclusion directly from the preceding CGWH base; no inheritance by arbitrary open subspaces is used. The complement $E^{(r)}$ is a vector subbundle: in a local frame for the ambient bundle the Hermitian orthogonal projection onto $L_r$ varies continuously, and projecting a basis of its kernel at a fixed point gives independent local sections on a neighborhood, by the nonvanishing of a minor. They span the kernel of this constant-rank projection there. Every such bundle is numerable, since the intermediate base is paracompact Hausdorff and Under choice and dependent choice, every open cover of a paracompact Hausdorff space admits a locally finite subordinate partition of unity supplies a partition on its linear chart cover (AC implies DC). This proves the hypotheses required at the next stage, by finite induction.

The $i$-th tautological line $L_i\to \mathrm{Fl}(E)$ is the sub-line bundle of $q^{\ast }E$ whose fiber over a flag $V_{\bullet }$ is $V_i\cap V_{i-1}^{\perp }$, the orthogonal complement of $V_{i-1}$ in $V_i$; it is a complex line bundle. Orthogonal decomposition of each flag gives a metric-preserving isomorphism of complex bundles $q^{\ast }E=L_1\oplus L_2\oplus \cdots \oplus L_n,$ where the summands are the tautological lines. The Chern roots of $E$ are the classes $t_i:=c_1(L_i)\in H^2(\mathrm{Fl}(E);\mathbb{Z})(1\le i\le n),$ defined by Chern classes from the projective-bundle relation after the Chern classes themselves exist; the summands and their first Chern classes are the data in which the splitting principle is stated.

For $n=1$ there are no projectivization steps: $\mathrm{Fl}(E)=B$, $q$ is the identity and $L_1=E$. Rank zero is outside this definition.