De Rham invariant

Short description: Mod 2 invariant of (4k+1)-dimensional manifold

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

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 [math]\displaystyle{ H_{2k}(M), }[/math] as an integer mod 2;
• the Stiefel–Whitney number [math]\displaystyle{ w_2w_{4k-1} }[/math];
• the (squared) Wu number, [math]\displaystyle{ v_{2k}Sq^1v_{2k}, }[/math] where [math]\displaystyle{ v_{2k} \in H^{2k}(M;Z_2) }[/math] is the Wu class of the normal bundle of [math]\displaystyle{ M }[/math] and [math]\displaystyle{ Sq^1 }[/math] is the Steenrod square ; formally, as with all characteristic numbers, this is evaluated on the fundamental class: [math]\displaystyle{ (v_{2k}Sq^1v_{2k},[M]) }[/math];
• in terms of a semicharacteristic.

References

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