Uniqueness of Chern classes from the splitting principle

Lines and rank zero. On every complex line over an allowed base, properties 2 and [F1] give $d(L)=1+e(L_{\mathbb{R}})=c(L)$; all terms of index at least two vanish. If $n=0$, both totals are 1 by their degree-zero and rank conventions, so the conclusion already holds. Henceforth take $n\ge 1$.

The flag space is path-connected. The base $B$ is admissible for [F2] by [F6]. In the tower of [F3], each fiber ${\mathbb{C}\mathbb{P}}^{m-1}$ is nonempty and path-connected: two distinct lines have linearly independent representatives $v,w$, and $[(1-t)v+tw]$ joins them, since that vector never vanishes; equal lines use the constant path. For a stage with path-connected base, join the images of two total-space points by a path. Lift it from the first point by [F4] and join its endpoint to the second point inside the terminal fiber. This proves that the stage total is path-connected. Finite induction proves that $F=\mathrm{Fl}(E)$ is nonempty and path-connected; rank one has $F=B$. By [F3], $F$ has CW type. No assertion that $F$ itself is a CW complex is made.

Replace the base before evaluating the assignment. Choose a homotopy equivalence $h:C\to F$ from a CW complex and a homotopy inverse $k:F\to C$, as supplied by CW type in step 1.2. The complex $C$ is nonempty and path-connected. Indeed, for $x,y\in C$, join $h(x)$ to $h(y)$ in $F$ and apply $k$; the homotopy $kh\simeq {\mathrm{id}}_C$ joins its endpoints to $x,y$. By [F5], $h^{\ast }$ is an isomorphism in integral cohomology, inverse to $k^{\ast }$. Set $p=qh:C\to B$ and $M_j=h^{\ast }{L}_j$. Pulling back the actual splitting of [F2] gives $p^{\ast }E\cong {⨁}_{j=1}^n{M}_j$. These bundles are numerable: pull back the locally finite partition on each original bundle chart cover; local finiteness and the identity sum are preserved by composition with $h$. Thus $d$ and [F1] both apply to the pulled-back bundles on the nonempty path-connected CW complex $C$. Also $p^{\ast }=h^{\ast }{q}^{\ast }$ is injective.

Compare on the CW model. By property 3, isomorphism invariance, and step 1.1 applied to the lines $M_j$ over $C$, $d(p^{\ast }E)={\prod }_{j=1}^{n}d(M_j)={\prod }_{j=1}^{n}c(M_j)=c(p^{\ast }E)\mathrm{.}$ The last equality is [F1]'s Whitney identity, applied on $C$, where its hypotheses hold. This proof never evaluates $d$ on a bundle over the merely CW-type space $F$.

Descend. The map $p:C\to B$ has both source and target in the stated assignment domain. Naturality of $d$ and [F1] gives $p^{\ast }(d(E)-c(E))=d(p^{\ast }E)-c(p^{\ast }E)=0$. The injectivity established in step 2.1 implies $d(E)=c(E)$, degree by degree. Conversely [F1] provides the Chern assignment satisfying all three properties, so this is uniqueness together with the already supplied existence.

Boundaries and conventions. Rank zero was settled before making a flag bundle. In rank one, normalization in step 1.1 suffices, and the tower is the identity. The empty base is explicitly excluded; no domain extension to disconnected or non-CW bases is claimed for $d$. Terms above the rank vanish, so each total and each product is finite even when the CW complexes are infinite-dimensional. The line normalization uses the complex orientation of $L_{\mathbb{R}}$, not an independent sign for a projective generator. AC is inherited through [A1]; the chosen single CW equivalence and the finite tower introduce no unrecorded family of choices.