De Rham invariant

The de Rham invariant of a (4k+1)-dimensional manifold can be defined in various equivalent ways:^[3]

• the rank of the 2-torsion in ${\displaystyle H_{2k}(M),}$ as an integer mod 2;
• the Stiefel–Whitney number ${\displaystyle w_{2}w_{4k-1}}$;
• the (squared) Wu number, ${\displaystyle v_{2k}Sq^{1}v_{2k},}$ where ${\displaystyle v_{2k}\in H^{2k}(M;Z_{2})}$ is the Wu class of the normal bundle of ${\displaystyle M}$ and ${\displaystyle Sq^{1}}$ is the Steenrod square; formally, as with all characteristic numbers, this is evaluated on the fundamental class: ${\displaystyle (v_{2k}Sq^{1}v_{2k},[M])}$;
• in terms of a semicharacteristic.