Framed cobordism (${\Omega }_{fr}^{\ast }(X)={\pi }_S^{\ast }(X)$)

Framed cobordism is the generalized cohomology theory of stable cohomotopy. Its geometric maps have a stable framing of a vector bundle on their normal data. A finite covering has canonically framed zero-dimensional relative tangent data; its Gysin class of $1$ is the stable transfer of the cover. A trivial $d$-sheeted cover gives $d\cdot 1$ in degree zero.

The degree-zero framed cobordism class is defined by the transfer of the subset cover of a framed vector bundle. With the canonical relative framing it equals $\left(\genfrac{}{}{0ex}{}{n}{k}\right)\cdot 1$ for a globally ordered framing of rank $n$. Put $l_0=1$ and $l_k=0$ outside the rank range. Disjoint-union additivity and fiber-product multiplicativity of transfer give $l_k(\eta \oplus \zeta )={\sum }_{i+j=k}{l}_i(\eta )l_j(\zeta )$.