Exotic subset-cover class of a framed bundle ($l_k(\eta )=(p_k{)}_{!}(1)$)

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 $l_k(\eta \oplus \zeta )={\sum }_{i+j=k}{l}_i(\eta )l_j(\zeta )$.