Total Stiefel–Whitney class of a sum of universal lines

The cohomology ring is polynomial. By [F2] applied to the $n$-fold product and [F1], the cross product identifies $H^{\ast }(({\mathbb{R}\mathbb{P}}^{\mathrm{\infty }}{)}^n;{\mathbb{F}}_2)$ with the $n$-fold graded tensor product of ${\mathbb{F}}_2[a]$, which is the polynomial ring ${\mathbb{F}}_2[a_1,\dots ,a_n]$ under the identifications $a_i={\mathrm{pr}}_i^{\ast }a$. The construction is by iterated Künneth over the finite product, and each factor's homology is ${\mathbb{F}}_2$ in each degree, so the finite-freeness hypothesis of [F2] holds factor by factor.

The total class of the sum. Each $L_i={\mathrm{pr}}_i^{\ast }{\gamma }_1$ is a line bundle, so naturality [F4] applied to the universal computation [F3] gives $w(L_i)={\mathrm{pr}}_i^{\ast }w({\gamma }_1)=1+{\mathrm{pr}}_i^{\ast }a=1+a_i\mathrm{.}$ The Whitney product formula [F4] applied to the finite sum gives $w(L_1\oplus \cdots \oplus L_n)={\prod }_{i=1}^n(1+a_i)$ in ${\mathbb{F}}_2[a_1,\dots ,a_n]$.

The individual classes. Expanding the product in the polynomial ring, the coefficient of degree $k$ is the sum of all products of $k$ distinct variables, that is the elementary symmetric polynomial ${\sigma }_k(a_1,\dots ,a_n)$; the coefficient of degree zero is $1=w_0$. Since the $a_i$ are algebraically independent by step 1.1, each ${\sigma }_k$ is nonzero for $k\le n$, which is the standard algebraic-independence input used to see that a rank-$n$ bundle can have all $n$ positive classes nonzero.

Boundary cases. For $n=0$ the product is a point, the empty sum of lines is the zero bundle, the empty product is $1$, and $w(0)=1$ by the rank-zero convention. For $n=1$ the statements reduce to $w({\gamma }_1)=1+a$ and ${\sigma }_1=a$. The polynomial ring is commutative, so the order of the factors does not matter, and the displayed class is independent of the chosen ordering of the coordinates. AC is used through Künneth and the Whitney formula, as recorded.