Mod-two real projective bundle theorem

The projection $P(E)\to B$ is a Serre fibration with fiber ${\mathbb{R}\mathbb{P}}^{n-1}$ whose classes $1,x,\dots ,x^{n-1}$ restrict to a basis of the fiber cohomology. This is [F1] combined with [F2]: the fiber over $b$ is $P(E_b)$ and the restrictions of the displayed classes are a basis.

Choose a homotopy equivalence $g:K\to B$ from a CW complex, put $E^{\mathrm{\prime }}=g^{\ast }E$, and form $g^{\mathrm{\prime }}:P(E^{\mathrm{\prime }})\to P(E)$. Projectivization commutes with this pullback: in a pulled-back linear chart the identification sends $(k,\mathrm{\ell })$ to $(k,(g(k),\mathrm{\ell }))$, and the formulas agree on overlaps since both have the same pulled-back linear transitions. Hence $P(E^{\mathrm{\prime }})\cong g^{\ast }P(E)$, with $g^{\mathrm{\prime }}$ restricting to the identity on each projective fiber. Both projective totals are paracompact Hausdorff of CW type by [F9]. Both projections are Serre fibrations by [F1]. In their natural homotopy sequences [F5], the fiber and base maps are isomorphisms on all homotopy groups. A direct exactness chase gives the same for the total map, including degree one: to lift a target total class, lift its base image by the base isomorphism; its fiber boundary vanishes by fiber injectivity, so it lifts to the source total group. Correct the difference using surjectivity on the fiber group. For injectivity, a source class killed in the target has zero base image, so comes from a fiber class. Its image is a target base boundary; lift that boundary through the base isomorphism and use fiber injectivity to see the original total class vanishes. The degree-one chase uses products rather than sums and the fact that the fiber is path connected, so the boundary to its component set is trivial. Path lifting with path-connected fibers identifies total components with base components. Thus $g^{\mathrm{\prime }}$ is a weak homotopy equivalence.

Suppose first that $B$ is a CW complex. Then [F3] applies to the Serre fibration of step 1.1 with the classes $e_i=x^i$, $i=0,\dots ,n-1$, which are homogeneous of degree $i$ and restrict to a basis by step 1.1, provided the base is path-connected. If $B$ is path-connected, the conclusion is that $\mathrm{\Phi }:{⨁}_{i=0}^{n-1}{H}^{\ast -i}(B;{\mathbb{F}}_2)⟶H^{\ast }(P(E);{\mathbb{F}}_2),(a_i)⟼{\sum }_{i=0}^{n-1}{p}^{\ast }{a}_i⌣x^i,$ is an $H^{\ast }(B;{\mathbb{F}}_2)$-module isomorphism, so $H^{\ast }(P(E);{\mathbb{F}}_2)$ is free over $H^{\ast }(B;{\mathbb{F}}_2)$ on $1,x,\dots ,x^{n-1}$. If $B$ is a general CW complex, its path components $B_{\alpha }$ are open and closed subcomplexes, because a CW complex is locally path-connected and cells are connected; restricting $E$ gives a numerable bundle over the path-connected CW complex $B_{\alpha }$, and $P(E)={\coprod }_{\alpha }P(E{\mathrm{\mid }}_{B_{\alpha }})$. For a disjoint union $X={\coprod }_{\alpha }{X}_{\alpha }$ of open and closed pieces the singular chain complex is the direct sum of the piece complexes, since the connected simplex ${\mathrm{\Delta }}^k$ maps into a single piece; dualising gives a product of cochain complexes, whose cycles are exactly the families of cycles and whose coboundaries are exactly the families of coboundaries, the latter using AC to select one primitive at each index. Hence $H^k(X;R)\cong {\prod }_{\alpha }{H}^k(X_{\alpha };R)$, and applying this to $B$ and to $P(E)$ turns the componentwise isomorphisms into the displayed $H^{\ast }(B;{\mathbb{F}}_2)$-module isomorphism.

By [F7], $g^{\mathrm{\prime }}$ is an integral homology isomorphism, and by [F6] and the module five lemma [F10], $g^{\mathrm{\prime }\ast }$ is an isomorphism on ${\mathbb{F}}_2$-cohomology. Also $g^{\ast }$ is a cohomology isomorphism by [F10]. Apply Leray–Hirsch over each CW component of $K$ using the specified classes $e_i=g^{\mathrm{\prime }\ast }(x^i)$, not an unproved identification with a separately defined tautological class on $P(E^{\mathrm{\prime }})$. They restrict to a fiber basis because $g^{\mathrm{\prime }}$ is the identity on each fiber and [F2] supplies that basis. The component argument of step 2.1 gives an isomorphism ${\mathrm{\Phi }}_K$ for these classes. Naturality [F4] gives $g^{\mathrm{\prime }\ast }{\mathrm{\Phi }}_B={\mathrm{\Phi }}_K({⨁}_i{g}^{\ast })$. The other three maps are isomorphisms, so ${\mathrm{\Phi }}_B$ is an isomorphism. Finite sums indexed by $0\le i<n$ commute with the degreewise component products. This proves the module claim on the stated general base.

Existence and uniqueness of the coefficients. By step 3.1 the elements $1,x,\dots ,x^{n-1}$ form a module basis of $H^{\ast }(P(E);{\mathbb{F}}_2)$ over $H^{\ast }(B;{\mathbb{F}}_2)$. The class $x^n\in H^n(P(E);{\mathbb{F}}_2)$ therefore has a unique expansion $x^n={\sum }_{i=0}^{n-1}{p}^{\ast }{b}_i⌣x^i$ with $b_i\in H^{n-i}(B;{\mathbb{F}}_2)$. Setting $c_i(E):=b_{n-i}\in H^i(B;{\mathbb{F}}_2)$, moving the terms to one side and using that the coefficient ring has characteristic two so that signs are trivial gives the unique relation $x^n+c_1(E)x^{n-1}+\cdots +c_n(E)=0$, whose leading coefficient is $1$.

The relation generates all relations. Let $\phi :H^{\ast }(B;{\mathbb{F}}_2)[x]\to H^{\ast }(P(E);{\mathbb{F}}_2)$ be the $H^{\ast }(B;{\mathbb{F}}_2)$-algebra homomorphism with $\phi (x)=x$ and $\phi (a)=p^{\ast }a$; it is well defined because [F4] makes $p^{\ast }$ a unital ring homomorphism and [F11] makes all classes commute. By step 3.1 it is surjective, since the module basis lies in its image. Let $f=x^n+c_1(E)x^{n-1}+\cdots +c_n(E)$ be the monic relation of step 4.1, of degree $n$. Division with remainder by a monic polynomial is available over any commutative ring, so every element of $H^{\ast }(B;{\mathbb{F}}_2)[x]$ has a unique representative ${\sum }_{i=0}^{n-1}{g}_i{x}^i$ modulo the principal ideal $(f)$, and the classes $1,x,\dots ,x^{n-1}$ are a module basis of $H^{\ast }(B;{\mathbb{F}}_2)[x]\mathrm{/}(f)$. The induced map $\bar{\phi }:H^{\ast }(B;{\mathbb{F}}_2)[x]\mathrm{/}(f)\to H^{\ast }(P(E);{\mathbb{F}}_2)$ sends that basis to the module basis of step 3.1 and is $H^{\ast }(B;{\mathbb{F}}_2)$-linear, so it is an isomorphism and $\ker \phi =(f)$.

Boundary cases. For $n=1$ the basis is $1$ alone and the relation is $x+c_1(E)=0$, so the argument above applies verbatim. For a disconnected base the degreewise identification of cohomology with the product over components was recorded in step 2.1 and transferred in step 3.1. If $B=\mathrm{\varnothing }$ then $P(E)=\mathrm{\varnothing }$ and all groups are zero, so the statement is valid with the unique $n$ zero coefficients. The one-point fiber ${\mathbb{R}\mathbb{P}}^0$ enters only through the basis statement for $n=1$. AC is used through [F1], [F3], the componentwise primitives of step 2.1 and the universal coefficient sequence of step 3.1, as recorded.