The total L-class and the L-genus of a smooth manifold

Assume AC, inherited from the Pontryagin-class construction (Pontryagin classes by complexification) and used only there and in the admissibility supplied by Smooth manifolds have CW homotopy type.

Let $M$ be a smooth manifold: finite-dimensional, Hausdorff and second countable, possibly with boundary, possibly disconnected, possibly empty. By Smooth manifolds have CW homotopy type such an $M$ is a CW-type base, its tangent bundle $TM$ is a numerable finite-rank real bundle, and the Pontryagin classes $p_i(TM)\in H^{4i}(M;\mathbb{Z})$ are defined for all $i\ge 0$, with $p_0=1$ and $p_i(TM)=0$ whenever $2i>\dim M$ (Pontryagin classes by complexification, Naturality, stability, and mod-two reduction of Pontryagin classes). The total L-class of $M$ is $L(M):=L(TM)\in {\widehat{H}}^{4\ast }(M;\mathbb{Q}),$ the total L-class of the tangent bundle (The Hirzebruch L-polynomials and the total L-class of a real vector bundle); its component of degree $4j$ is $L_j(TM)=L_j(p_1(TM),\dots ,p_j(TM))$, and for disconnected $M$ the class is taken componentwise. Naturality and stability on CW-type bases are supplied by the transport argument of Pontryagin numbers of a closed oriented manifold, and the L-identities extend to these bases by The L-polynomials are well defined and form a multiplicative, natural and stable sequence. The construction is well defined because each component is a polynomial in the Pontryagin classes and the completed ring is the product of the groups $H^{4j}(M;\mathbb{Q})$ (The completed cohomology ring in degrees divisible by four, The completed fourfold-graded cohomology ring is natural and satisfies the ring laws).

The L-genus. Let $M$ be a closed oriented smooth manifold of dimension $4k$, $k\ge 0$, with fundamental class $[M]\in H_{4k}(M;\mathbb{Z})$. The L-genus of $M$ is the rational characteristic number $L[M]:=⟨L_k(TM),[M]⟩\in \mathbb{Q},$ the degree-$4k$ evaluation of the total L-class under the Kronecker pairing (Kronecker evaluation pairing, Fundamental class of a compact oriented manifold). Expanding the weight-$4k$ polynomial $L_k={\sum }_{\mathrm{\mid }J\mathrm{\mid }=k}{c}_J{p}_J$ with $c_J\in \mathbb{Q}$ and $p_J=p_{j_1}\cdots p_{j_r}$ (The Hirzebruch L-polynomials and the total L-class of a real vector bundle) gives $L[M]={\sum }_{\mathrm{\mid }J\mathrm{\mid }=k}{c}_J{p}_J[M],$ so the L-genus is a rational linear combination of the Pontryagin numbers $p_J[M]$ of Pontryagin numbers of a closed oriented manifold.

Zero extension. Since $L(M)={\sum }_{j\ge 0}{L}_j(TM)$ is concentrated in degrees divisible by four, a closed oriented manifold whose dimension is not divisible by four has no degree-$4j$ component in its dimension and no L-genus of the above form; one declares $L[M]:=0$ in that case as a bookkeeping extension only (see the closing bookkeeping remark on this page). Naturality and multiplicativity of $L$ are properties proved in The L-polynomials are well defined and form a multiplicative, natural and stable sequence, not part of this definition.