The first Stiefel–Whitney class classifies orientability

The model ${\mathbb{R}\mathbb{P}}^{\mathrm{\infty }}$ is a $K(\mathbb{Z}\mathrm{/}2,1)$. By [F6] the antipodal quotient $q:S^{\mathrm{\infty }}\to {\mathbb{R}\mathbb{P}}^{\mathrm{\infty }}$ is numerable: its local covering charts admit a numeration on the paracompact CW base. Thus [F11] makes it a Serre fibration with fiber $S^0$ and contractible total space, so the exact sequence of [F6] gives ${\pi }_i({\mathbb{R}\mathbb{P}}^{\mathrm{\infty }})\cong {\pi }_{i-1}(S^0)=0$ for $i\ge 2$. In degree one, use the action clause of the same fibration theorem rather than treating pointed-set exactness as injectivity: ${\pi }_1({\mathbb{R}\mathbb{P}}^{\mathrm{\infty }})$ acts on the two components of $S^0$, its orbits are the fibers of ${\pi }_0(S^0)\to {\pi }_0(S^{\mathrm{\infty }})$ and hence form one transitive orbit, and the stabilizer of the chosen component is the image of ${\pi }_1(S^{\mathrm{\infty }})=0$. Therefore the orbit map from ${\pi }_1({\mathbb{R}\mathbb{P}}^{\mathrm{\infty }})$ to the two-point set ${\pi }_0(S^0)$ is bijective, so the fundamental group has two elements and is $\mathbb{Z}\mathrm{/}2$. The base is path connected as the image of the contractible total space. Since ${\mathbb{R}\mathbb{P}}^{\mathrm{\infty }}$ is a based CW complex whose only nonzero homotopy group is this one, it is a model of $K(\mathbb{Z}\mathrm{/}2,1)$ in the sense of [F1], and the representability theorem [F1] applies to it.

Tensor products add over a CW complex. The ordinary product of the two countable CW complexes ${\mathbb{R}\mathbb{P}}^{\mathrm{\infty }}$ is a CW complex (Hatcher, Algebraic Topology, Appendix Theorem A.6, printed p.524), hence an admissible base. By [F10] each homology group of ${\mathbb{R}\mathbb{P}}^{\mathrm{\infty }}$ has one-dimensional dual by [F7], hence is itself one-dimensional: two independent vectors would extend to a basis and give two independent coordinate functionals under AC, whereas the zero space has zero dual. Thus the finite-free hypothesis in [F7] holds. Let $q_1,q_2:{\mathbb{R}\mathbb{P}}^{\mathrm{\infty }}\times {\mathbb{R}\mathbb{P}}^{\mathrm{\infty }}\to {\mathbb{R}\mathbb{P}}^{\mathrm{\infty }}$ be the projections and put $N=q_1^{\ast }{\gamma }_1\otimes q_2^{\ast }{\gamma }_1$, a numerable real line bundle. By [F7] every class of $H^1({\mathbb{R}\mathbb{P}}^{\mathrm{\infty }}\times {\mathbb{R}\mathbb{P}}^{\mathrm{\infty }};{\mathbb{F}}_2)$ is uniquely $\alpha a_1+\beta a_2$ with $\alpha ,\beta \in {\mathbb{F}}_2$, the two axis inclusions returning $\alpha a$ and $\beta a$. Let $c_N$ classify $N$, so that $w_1(N)=c_N^{\ast }a$ by [F2]. The composite $q_1{i}_1$ is the identity and $q_2{i}_1$ is constant, and a pullback along a constant map is a trivial line bundle by its fiber description, so $i_1^{\ast }N\cong {\gamma }_1\otimes {\epsilon }^1\cong {\gamma }_1$; naturality [F4] therefore gives $i_1^{\ast }{w}_1(N)=w_1(i_1^{\ast }N)=w_1({\gamma }_1)=a$, so the coefficient of $a_1$ is one, and symmetrically that of $a_2$ is one: $w_1(N)=a_1+a_2$. Now let $L,M$ be numerable real line bundles over a CW complex $B$, classified by maps $c_L,c_M:B\to {\mathbb{R}\mathbb{P}}^{\mathrm{\infty }}$, so that $L\cong c_L^{\ast }{\gamma }_1$ and $M\cong c_M^{\ast }{\gamma }_1$ by [F2]. Tensor products commute with pullback [F3], so $L\otimes M\cong (c_L,c_M{)}^{\ast }N$, and naturality [F4] together with the class of $N$ gives $w_1(L\otimes M)=(c_L,c_M{)}^{\ast }(a_1+a_2)=c_L^{\ast }a+c_M^{\ast }a=w_1(L)+w_1(M)\mathrm{.}$ Since ${\mathrm{\Lambda }}^2(L\oplus M)\cong L\otimes M$ by [F3] while $w_1(L\oplus M)=w_1(L)+w_1(M)$ by the Whitney formula [F4], this also gives $w_1(\det (L\oplus M))=w_1(L\oplus M)$ for the rank-two sum.

Line bundles over a CW complex are classified by $w_1$. First let $B$ be connected with vertex $b_0$. Step 1.1 and [F1] give a bijection $[B,{\mathbb{R}\mathbb{P}}^{\mathrm{\infty }}{]}_{\ast }\to H^1(B;{\mathbb{F}}_2)$ by pulling back the fundamental class. That class is nonzero: apply [F1] to the model itself, whose identity cannot be based nullhomotopic because it induces the identity on its nonzero fundamental group. It is therefore the unique nonzero class $a$ of [F7]. Every unbased map can be made based: choose a path from its value at $b_0$ to the target vertex and extend this vertex homotopy using [F9]. If two based maps are freely homotopic, their pullbacks of $a$ agree by [F8], so injectivity of [F1] already makes them based homotopic. Thus forgetting basepoints is a bijection. Compose this proved unbased bijection with [F2]; its value on the bundle classified by $c$ is $c^{\ast }a=w_1(L)$. Both bijections are natural in the base, as is $w_1$ by [F4], and over a disconnected CW complex both sides split as products over the components, since a line bundle, a classifying map and a cohomology class are each determined componentwise. In particular the trivial bundle corresponds to $0$, so $w_1(L)=0$ forces $L$ to be trivial.

Admissible bases by transfer along a CW model. Let $B$ be admissible and choose a homotopy equivalence $g:K\to B$ from a CW complex $K$ with homotopy inverse $h$, which exists by the definition of CW homotopy type and [F8]. Pullback along $g$ is a bijection ${\mathrm{Vect}}_1^{\mathbb{R}}(B)\to {\mathrm{Vect}}_1^{\mathbb{R}}(K)$, and $g^{\ast }:H^1(B;{\mathbb{F}}_2)\to H^1(K;{\mathbb{F}}_2)$ is an isomorphism, both by [F8]; naturality of $w_1$ [F4] gives $w_1(g^{\ast }L)=g^{\ast }{w}_1(L)$ for every numerable real line bundle $L$ over $B$, so the square comparing the two bases commutes. Over the CW complex $K$ the corresponding map is a bijection by step 2.1, and in a commuting square whose other three maps are bijections the fourth map is a bijection as well; hence $L\mapsto w_1(L)$ is a bijection over $B$. The tensor identity transfers the same way: tensor products commute with pullback [F3] and step 1.2 applies over the CW complex $K$, so $g^{\ast }{w}_1(L\otimes M)=w_1(g^{\ast }(L\otimes M))=w_1(g^{\ast }L\otimes g^{\ast }M)=w_1(g^{\ast }L)+w_1(g^{\ast }M)=g^{\ast }(w_1(L)+w_1(M)),$ and injectivity of $g^{\ast }$ [F8] gives $w_1(L\otimes M)=w_1(L)+w_1(M)$ over $B$.

The first class is the class of the determinant line. Let $E\to B$ have rank $n\ge 1$ over the admissible base $B$ and let $q:\mathrm{Fl}(E)\to B$ be its flag bundle, so $q^{\ast }E\cong L_1\oplus \cdots \oplus L_n$ and $q^{\ast }$ is injective. By [F4] and the tensor identity of step 3.1, applied over the admissible base $\mathrm{Fl}(E)$, $w_1(q^{\ast }E)={\sum }_{j=1}^n{w}_1(L_j)=w_1(L_1\otimes \cdots \otimes L_n)=w_1(q^{\ast }\det E)=q^{\ast }{w}_1(\det E),$ because $\det (q^{\ast }E)\cong q^{\ast }\det E$ and $w_1$ is natural [F4]. Injectivity of $q^{\ast }$ gives $w_1(E)=w_1(\det E)$.

Orientability and the determinant line. Supply $E$ with the metric of [F3]. An orientation of $E_b$ is a choice of generator of ${\mathrm{\Lambda }}^n{E}_b$ up to positive scaling, so the orientation cover of $E$ is identified fiberwise with the unit sphere bundle $S(\det E)$ of the determinant line, the map sending an orientation to its unit volume element being a homeomorphism over $B$: in orthonormal frames it identifies the two signs, and both transition rules are multiplication by the determinant sign. Hence $E$ is orientable exactly when $S(\det E)$ admits a section, which happens exactly when the line bundle $\det E$ is trivial, since a nowhere-zero section of a line bundle trivializes it and conversely. By the bijection of step 3.1 over the admissible base $B$, the determinant line is trivial exactly when $w_1(\det E)=0$, which by step 4.1 is exactly $w_1(E)=0$. The equivalence with an $\mathrm{SO}(n)$-reduction of the orthonormal frame bundle is [F5].

Boundary cases. For $n=0$ the bundle has its canonical orientation by [F5], the determinant line is the trivial line, and $w_1(E)=0$ by the rank convention, so both sides of the equivalence hold. For $n=1$ the determinant line is $E$ itself and step 4.1 is the identity, while step 3.1 is the asserted classification of line bundles. For a trivial bundle of positive rank, the wedge of its standard frame is a nowhere-zero section of its determinant line; steps 3.1 and 4.1 then give $w_1=0$. If $B=\mathrm{\varnothing }$ all groups are zero and the unique empty bundle is orientable, matching $w_1=0$. AC is inherited through representability, classification, metrics, splitting, Kunneth, duality, fibration and homotopy-invariance interfaces.