De Rham invariant

In geometric topology, the de Rham invariant is a mod 2 invariant of a (4k+1)-dimensional manifold, that is, an element of $\mathbf {Z} /2$ – either 0 or 1. It can be thought of as the simply-connected symmetric L-group $L^{4k+1},$ and thus analogous to the other invariants from L-theory: the signature, a 4k-dimensional invariant (either symmetric or quadratic, $L^{4k}\cong L_{4k}$), and the Kervaire invariant, a (4k+2)-dimensional quadratic invariant $L_{4k+2}.$

It is named for Swiss mathematician Georges de Rham, and used in surgery theory.

Definition

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

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

References

• Lusztig, G.; Milnor, J.; Peterson, F. P. (1969), "Semi-characteristics and cobordism", Topology, 8: 357–360, doi:10.1016/0040-9383(69)90021-4
• Chess, Daniel, A Poincaré-Hopf type theorem for the de Rham invariant, 1980