Integral complex projective bundle theorem

Since $E$ is numerable, choose a subordinate partition of unity on a vector-bundle trivializing cover. By [F1] that same cover and the same partition trivialize and numerate $P(E)\to B$, whose fiber ${\mathbb{C}\mathbb{P}}^{n-1}$ is compact Hausdorff. Thus [F4] makes $p$ a Hurewicz, hence Serre, fibration over the path-connected CW complex $B$.

By [F2] the classes $1,x_R,\dots ,x_R^{n-1}$ restrict on every fiber to an $R$-basis of $H^{\ast }({\mathbb{C}\mathbb{P}}^{n-1};R)$: for $R=\mathbb{Z}$ directly, and for $R={\mathbb{F}}_p$ through the coefficient reductions.

Construction on bases of CW type. Let $B^{\mathrm{\prime }}$ be paracompact Hausdorff CGWH of CW type. Projectivizing the given linear charts and their transitions produces a fiber bundle with the same numeration, by exactly the quotient-chart maps in [F1]. By [F5] its total space $T=P(E)$ is paracompact Hausdorff, CGWH, and of CW type. On each projective coordinate domain $v_j\mathrm{\ne }0$, the representative with $v_j=1$ trivializes the tautological line; [F7] numerates this chart cover. The real frames $(v,iv)$ agree in orientation because multiplication by $a+ib\mathrm{\ne }0$ has determinant $a^2+b^2>0$. Thus the real rank-two tautological bundle is oriented, numerable and in Thom scope, so [F7] defines $x$ on $T$. Its restriction on each fiber is the tautological Euler class by oriented naturality, and applying [F2] to the trivial rank-$n$ bundle over a point gives the required integral and prime-field fiber bases.

Choose a homotopy equivalence $w:W\to B^{\mathrm{\prime }}$ with $W$ a CW complex and form $T_W=w^{\ast }T$ with projection $q:T_W\to T$. Both bundle projections are Serre fibrations by [F4] and [F8]. In their natural homotopy sequences, the fiber map is the identity on ${\mathbb{C}\mathbb{P}}^{n-1}$ and the base maps induce isomorphisms. Hence $q$ induces isomorphisms on every positive homotopy group. Explicitly, for surjectivity of the middle map, lift a base class through the base isomorphism; its boundary vanishes by injectivity on the fiber group, so exactness lifts it to the source total group. Correct the difference from the target class using surjectivity on the fiber group. For injectivity, an element killed in the target has zero base image, hence comes from a fiber element. That fiber element maps to a base boundary in the target; lift that boundary class through the base isomorphism and use injectivity on the fiber group to conclude that the original element is zero. This group argument also works in degree one with multiplication in place of addition: the fiber is path connected, so both boundary maps to its component set vanish. Path lifting and path-connected fibers identify the components of each total space with those of its base, giving a bijection on components as well. Thus $q$ is a weak homotopy equivalence. By [F9], $q^{\ast }$ is a cohomology isomorphism with coefficients $\mathbb{Z}$ or ${\mathbb{F}}_p$; its ring structure is preserved by pullback.

Applying [F3] to the fibration of step 1.1 with the classes of step 1.2 gives the $H^{\ast }(B;R)$-module isomorphism ${⨁}_{i=0}^{n-1}{H}^{\ast -2i}(B;R)\to H^{\ast }(P(E);R)$ sending $(a_i)$ to ${\sum }_i{p}^{\ast }{a}_i⌣x_R^i$. In particular $1,x_R,\dots ,x_R^{n-1}$ are a basis of the free module $H^{\ast }(P(E);R)$ over $H^{\ast }(B;R)$.

The monic relation. Apply step 2.1 with $R=\mathbb{Z}$ to the element $x^n\in H^{2n}(P(E);\mathbb{Z})$: there are unique classes $b_i\in H^{2n-2i}(B;\mathbb{Z})$, $0\le i\le n-1$, with $x^n={\sum }_{i=0}^{n-1}{b}_i{x}^i$; setting $a_j:=(-1{)}^{j+1}{b}_{n-j}$ for $1\le j\le n$ turns this into $x^n-a_1{x}^{n-1}+a_2{x}^{n-2}-\cdots +(-1{)}^n{a}_n=0$, with $a_j\in H^{2j}(B;\mathbb{Z})$ by the grading. Uniqueness of the $a_j$ is uniqueness of the coefficients $b_i$ in the basis of step 2.1.

All relations. Let $f(t)=t^n-a_1{t}^{n-1}+\cdots +(-1{)}^n{a}_n\in H^{\ast }(B;\mathbb{Z})[t]$ and let $P\in H^{\ast }(B;\mathbb{Z})[t]$ satisfy $P(x)=0$. All coefficients of $f$ and the degree-two variable are central by [F10]. Successively subtracting the leading coefficient times the appropriate power of $t$ times $f$ reduces the degree, over this possibly noncommutative coefficient ring. Thus monic division gives $P=Qf+R$ with $\mathrm{deg}R<n$, and evaluating at $x$ gives $R(x)=0$, say $R(t)={\sum }_{i<n}{r}_i{t}^i$ with $r_i\in H^{\ast }(B;\mathbb{Z})$; then ${\sum }_i{p}^{\ast }{r}_i⌣x^i=0$ in $H^{\ast }(P(E);\mathbb{Z})$. By the basis property of step 2.1 all $r_i=0$, so $R=0$ and $P=Qf$ lies in the ideal generated by $f$.

Apply [F3] on each CW component of $W$ with the classes $q^{\ast }(x_R^i)$. Their restrictions form the bases proved in step 1.3; no Euler construction on $W$ is needed here. The natural square of cup-product maps has vertical isomorphisms $w^{\ast }$ (by [F6]) and $q^{\ast }$ (by step 1.4), so the module map for $B^{\mathrm{\prime }}$ is an isomorphism. For disconnected $W$, singular cohomology is the product of component cohomologies in each degree: singular simplices lie in a single component. The finite direct sum indexed by $0\le i<n$ commutes with this product. Therefore the same module map is an isomorphism without connectedness. Steps 3.1 and 4.1 apply to this module map and give the monic relation and its full relation ideal on $B^{\mathrm{\prime }}$.

Boundary cases. For $n\ge 1$ the basis $1,x,\dots ,x^{n-1}$ is nonempty and begins with the unit $1$; in the rank-one case $n=1$ the module is $H^{\ast }(B;R)$ itself and the relation reads $x-a_1=0$, so $a_1=x$ and no higher $a_i$ occurs. The zero bundle is excluded by $n\ge 1$, an empty base gives the unique zero cohomology groups; a disconnected base is handled by step 5.1, and the coefficient rings $\mathbb{Z}$ and ${\mathbb{F}}_p$ are nonzero by hypothesis. The relation has leading term $x^n$ and constant term $(-1{)}^n{a}_n$, with all intermediate coefficients verified in step 3.1; the ring $H^{\ast }(B;\mathbb{Z})[t]$ admits monic division by the leading-term subtraction of step 4.1 with central even coefficients, so no domain hypothesis is used. AC is inherited through [A1] in the numerable-fibration, Thom, partition, Leray–Hirsch and coefficient suppliers.