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 – either 0 or 1. It can be thought of as the simply-connected symmetric L-group and thus analogous to the other invariants from L-theory: the signature, a 4k-dimensional invariant (either symmetric or quadratic, ), and the Kervaire invariant, a (4k+2)-dimensional quadratic invariant
The de Rham invariant of a (4k+1)-dimensional manifold can be defined in various equivalent ways:
- the rank of the 2-torsion in as an integer mod 2;
- the Stiefel–Whitney number ;
- the (squared) Wu number, where is the Wu class of the normal bundle of and is the Steenrod square ; formally, as with all characteristic numbers, this is evaluated on the fundamental class: ;
- in terms of a semicharacteristic.