Stiefel-Whitney classes of the tangent bundle of real projective space

The affine charts make ${\mathbb{R}\mathbb{P}}^m$ a boundaryless smooth manifold. It is compact: every line has a unit representative, and the quotient projection $S^m\to {\mathbb{R}\mathbb{P}}^m$ is continuous and surjective; $S^m$ is closed and bounded, hence compact, and its image is compact by [F5]. Fix a line $\mathrm{\ell }$ and a complement $H$, and let $p:{\mathbb{R}}^{m+1}\to \mathrm{\ell }$ be projection along $H$. The lines transverse to $H$ form an open set $U_{\mathrm{\ell }}$: on every affine chart, transversality is the nonvanishing of a linear coordinate expression. Each such line is uniquely the graph of $f\in \mathrm{Hom}(\mathrm{\ell },H)$. In affine coordinates this graph chart and its inverse are ratios of linear expressions with nonzero denominators, so are smooth by [F1] and [F2]. Differentiating at the graph of $0$ and composing $H\cong {\mathbb{R}}^{m+1}\mathrm{/}\mathrm{\ell }$ gives an isomorphism ${\mathrm{\Theta }}_{\mathrm{\ell }}:T_{\mathrm{\ell }}{\mathbb{R}\mathbb{P}}^m\to \mathrm{Hom}(\mathrm{\ell },{\mathbb{R}}^{m+1}\mathrm{/}\mathrm{\ell })$.

This tangent identification is independent of $H$. For a second complement $H^{\mathrm{\prime }}$, write $p^{\mathrm{\prime }},q^{\mathrm{\prime }}$ for the projections onto $\mathrm{\ell },H^{\mathrm{\prime }}$. Near $f=0$ the graph transition is $f^{\mathrm{\prime }}=q^{\mathrm{\prime }}f({\mathrm{id}}_{\mathrm{\ell }}+p^{\mathrm{\prime }}f{)}^{-1}\mathrm{.}$ Indeed, a graph vector $u+f(u)$ has $\mathrm{\ell }$-coordinate $({\mathrm{id}}_{\mathrm{\ell }}+p^{\mathrm{\prime }}f)u$ and $H^{\mathrm{\prime }}$-coordinate $q^{\mathrm{\prime }}f(u)$. The derivative of this transition at $0$ is $g\mapsto q^{\mathrm{\prime }}g$, since $f=0$ there; modulo $\mathrm{\ell }$, $q^{\mathrm{\prime }}g(u)$ and $g(u)$ agree. Thus both differentials give the same ${\mathrm{\Theta }}_{\mathrm{\ell }}$. These maps define a fibrewise isomorphism $\mathrm{\Theta }:T{\mathbb{R}\mathbb{P}}^m\to \mathrm{Hom}(L,{\epsilon }^{m+1}\mathrm{/}L)$, with the quotient and Hom bundles supplied by [F3] and [F4].

The map $\mathrm{\Theta }$ is a smooth bundle isomorphism. Over $U_{\mathrm{\ell }}$, the chart differential of ${\phi }_{\mathrm{\ell }}$ trivializes $T{\mathbb{R}\mathbb{P}}^m{\mathrm{\mid }}_{U_{\mathrm{\ell }}}\cong U_{\mathrm{\ell }}\times \mathrm{Hom}(\mathrm{\ell },H)$, and the same graph data trivialize $\mathrm{Hom}(L,{\epsilon }^{m+1}\mathrm{/}L){\mathrm{\mid }}_{U_{\mathrm{\ell }}}$: at ${\mathrm{\ell }}^{\mathrm{\prime }}=\mathrm{\Gamma }(f)$ the projection ${\mathbb{R}}^{m+1}\to \mathrm{\ell }$ along $H$ restricts to a linear isomorphism $L_{{\mathrm{\ell }}^{\mathrm{\prime }}}\to \mathrm{\ell }$, while $u+g\mapsto g-f(u)$ ($u\in \mathrm{\ell }$, $g\in H$) is a linear map killing $L_{{\mathrm{\ell }}^{\mathrm{\prime }}}$ and inducing an isomorphism ${\mathbb{R}}^{m+1}\mathrm{/}{L}_{{\mathrm{\ell }}^{\mathrm{\prime }}}\to H$. Both depend polynomially on $f={\phi }_{\mathrm{\ell }}({\mathrm{\ell }}^{\mathrm{\prime }})$, hence smoothly on ${\mathrm{\ell }}^{\mathrm{\prime }}$, and relative to these two trivializations $\mathrm{\Theta }$ is the identity map of $U_{\mathrm{\ell }}\times \mathrm{Hom}(\mathrm{\ell },H)$: the derivative of the straight slope curve $f+tg$ is $g$, whose image under the graph trivialization of the Hom-bundle is again $g$ by the formula just displayed. A map that is the identity in local trivializations is smooth, and $\mathrm{\Theta }$ is bijective with fibrewise-linear inverse, so it is a smooth bundle isomorphism $T{\mathbb{R}\mathbb{P}}^m\cong L^{\ast }\otimes ({\epsilon }^{m+1}\mathrm{/}L)$.

By [F4] the Euclidean metric gives smooth isomorphisms ${\epsilon }^{m+1}=L\oplus L^{\perp }$, $L^{\perp }\cong {\epsilon }^{m+1}\mathrm{/}L$ and $L^{\ast }\cong L$. Also $\mathrm{Hom}(L,L)$ is canonically trivial, with the identity as a nowhere-zero section. Tensoring the splitting with $L^{\ast }$ and using step 3.1 yields $T{\mathbb{R}\mathbb{P}}^m\oplus {\epsilon }^1\cong L^{\ast }\otimes ({\epsilon }^{m+1}\mathrm{/}L)\oplus L^{\ast }\otimes L\cong L^{\ast }\otimes {\epsilon }^{m+1}\cong (m+1)L^{\ast }\cong (m+1)L\mathrm{.}$ All these isomorphisms are smooth; no continuous metric is substituted for a smooth one.

Finally $\mathrm{\Theta }$ and the splitting are used to compute the classes. The bundle ${\epsilon }^{m+1}$ over the one-point base has $P({\epsilon }^{m+1})={\mathbb{R}\mathbb{P}}^m$ and tautological line ${\gamma }_{{\epsilon }^{m+1}}=L$, so by [F7] its tautological class is $x_L=x_{{\epsilon }^{m+1}}\in H^1({\mathbb{R}\mathbb{P}}^m;{\mathbb{F}}_2)$ and the projective-bundle relation is $x_L^{m+1}=0$ (all $c_i\in H^{>0}(\mathrm{p}\mathrm{t};{\mathbb{F}}_2)$ vanish), while $1,x_L,\dots ,x_L^m$ is a basis; since $H^1({\mathbb{R}\mathbb{P}}^m;{\mathbb{F}}_2)={\mathbb{F}}_{2}a$ has the unique nonzero class $a$ and a basis element cannot be zero, $x_L=a$. Hence $w(L)=1+x_L=1+a$ by the rank-one case of [F6]. Applying [F6] to the stable isomorphism of step 4.1, $w(T{\mathbb{R}\mathbb{P}}^m)=w(T{\mathbb{R}\mathbb{P}}^m\oplus {\epsilon }^1)=w((m+1)L)=w(L{)}^{m+1}=(1+a{)}^{m+1}$ in $H^{\ast }({\mathbb{R}\mathbb{P}}^m;{\mathbb{F}}_2)={\mathbb{F}}_2[a]\mathrm{/}(a^{m+1})$: the first equality is the stability clause for trivial summands, the second is invariance of the classes under bundle isomorphisms, and the third is the Whitney product formula iterated over the $(m+1)$ summands.