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

It is named for Swiss mathematician Georges de Rham, and used in surgery theory.[1][2]

Definition

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.

References

1. Morgan, John W; Sullivan, Dennis P. (1974), "The transversality characteristic class and linking cycles in surgery theory", Annals of Mathematics, 2, 99 (3): 463–544, doi:10.2307/1971060, JSTOR 1971060, MR 0350748
2. John W. Morgan, A product formula for surgery obstructions, 1978
3. (Lusztig, Milnor & Peterson 1969)

• Lusztig, George; Milnor, John; Peterson, Franklin P. (1969), "Semi-characteristics and cobordism", Topology, 8 (4): 357–360, doi:10.1016/0040-9383(69)90021-4, MR 0246308
• Chess, Daniel, A Poincaré-Hopf type theorem for the de Rham invariant, 1980