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 $$w_2w_{4k-1}$$;
 * the (squared) Wu number, $$v_{2k}Sq^1v_{2k},$$ 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: $$(v_{2k}Sq^1v_{2k},[M])$$;
 * in terms of a semicharacteristic.