Inertial and bordism properties of spheres

Abstract:

The $k$-connective bounding group ${\theta ^n}(k)$ and the $k$-connective inertial group ${I^n}(k)$ are defined as subgroups of ${\theta ^n}$, the group of smooth $n$-spheres, $n \geqq 7$. It is shown ${I^n}(k)$ is contained in ${\theta ^n}(k)$. Consequently, the image of the Milnor-Novikov pairing ${\tau _{n,k}}$ is contained in ${\theta ^{n + k}}(k)$ when $n \geqq k + 2$. It follows that ${\tau _{7,3}} = 0$.