The Hirzebruch L-polynomials and the total L-class of a real vector bundle

Work over $\mathbb{Q}$ with the series $Q(x)=x\mathrm{/}\tanh x={\sum }_{j\ge 0}{q}_{2j}{x}^{2j}$ of The formal hyperbolic tangent series and the even series $x\mathrm{/}\tanh x$, so that $q_0=1$ and, as recorded and justified in that definition, $q_2=1\mathrm{/}3$ and $q_4=-1\mathrm{/}45$. Give the $i$-th polynomial variable weight $4i$.

The L-polynomials. For each $k\ge 0$ let $L_k\in \mathbb{Q}[p_1,p_2,p_3,\dots ]$ denote the polynomial, homogeneous of weight $4k$, characterized by the following condition: for every $N\ge 1$ and all indeterminates $x_1,\dots ,x_N$, with $e_j$ the elementary symmetric polynomials (The elementary symmetric polynomials $e_0,e_1,\dots ,e_n$, Symmetric polynomials as the invariants of variable permutations), $L_k(e_1(x_1^2,\dots ,x_N^2),\dots ,e_k(x_1^2,\dots ,x_N^2))=[\text{weight }4k]{\prod }_{i=1}^{N}Q(x_i)\mathrm{.}$ Here $[\text{weight }4k]$ selects the homogeneous component of weight $4k$.

Existence and uniqueness. Give each $x_i$ weight $2$ and put $u_i=x_i^2$, of weight $4$. For $N\ge \max (1,k)$, the weight-$4k$ component of ${\prod }_{i}Q(x_i)$ is a symmetric polynomial of ordinary degree $k$ in the $u_i$. The fundamental theorem of symmetric polynomials (Fundamental theorem of symmetric polynomials: unique expression as a polynomial in $e_1,\dots ,e_n$) expresses it uniquely in $e_1(u),\dots ,e_N(u)$. Since $e_j$ has ordinary degree $j$, uniqueness of homogeneous components makes this expression homogeneous of weighted degree $k$ and excludes every $e_j$ with $j>k$. Define $L_k$ using $N=\max (1,k)$. Setting additional roots to zero leaves the product unchanged because $Q(0)=1$; injectivity of substitution in at least $k$ variables shows that the same polynomial works for every larger $N$. Setting roots to zero then gives the identity for $N<k$ too, where $e_j=0$ for $j>N$; injectivity is asserted only when $N\ge k$. In particular $L_0=1$. For two variables the weight-$4$ part of $Q(x_1)Q(x_2)$ is $q_2(x_1^2+x_2^2)=q_2{e}_1$, and its weight-$8$ part is $q_4(x_1^4+x_2^4)+q_2^2{x}_1^2{x}_2^2$, which the identities $x_1^4+x_2^4=e_1^2-2e_2$ and $x_1^2{x}_2^2=e_2$ rewrite as $q_4(e_1^2-2e_2)+q_2^2{e}_2$; substituting $q_2=1\mathrm{/}3$, $q_4=-1\mathrm{/}45$ and the corresponding powers of $p_1,p_2$ gives $L_1=\frac{p_1}{3},L_2=\frac{7p_2-p_1^2}{45}\mathrm{.}$

The total L-class. Assume AC (The Axiom of Choice), inherited from the Pontryagin and Chern constructions, including their bundle and Thom suppliers; no claim is made that DC alone supplies those constructions. For a real vector bundle $E\to B$ of finite rank over a CW-type base with Pontryagin classes $p_i(E)\in H^{4i}(B;\mathbb{Z})$ of Pontryagin classes by complexification, regarded in rational cohomology, the total L-class is the element $L(E):=(L_k(p_1(E),\dots ,p_k(E)){)}_{k\ge 0}\in {\widehat{H}}^{4\ast }(B;\mathbb{Q})$ of the completed ring of The completed cohomology ring in degrees divisible by four; its degree-$4k$ component is written $L_k(E):=L_k(p_1(E),\dots ,p_k(E))$. The sequence is well defined because each $L_k(p_1(E),\dots ,p_k(E))$ is a class in $H^{4k}(B;\mathbb{Q})$ computed from the given Pontryagin classes, and $p_i(E)=0$ whenever $2i>\mathrm{rank}E$ (Naturality, stability, and mod-two reduction of Pontryagin classes), and if $B$ has a finite-dimensional CW model, its cohomology vanishes above that dimension, so only finitely many components are nonzero. The coefficients $q_{2j}$ belong to $\mathbb{Q}$, so no integrality of $L(E)$ is asserted; see Formal power series over a commutative ring and the coefficient-extraction functional $[x^n]$ for the coefficient notation. Naturality, stability and multiplicativity of $L$ are not part of this definition; they are proved in the lemma named in justified_by.