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, and the Kervaire invariant, a (4k+2)-dimensional invariant.

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 H_2k(M), as an integer mod 2;
• the Stiefel–Whitney number w₂w_{4k − 1};
• the (squared) Wu number, v_2ksq¹v_2k, where v_2k is the Wu class and sq¹ is the Steenrod square; formally, as with all characteristic numbers, this is evaluated on the fundamental class: (v_2ksq¹v_2k(M),[M]);
• in terms of a semicharacteristic.

References

1. ^ Morgan & Sullivan, The transversality characteristic class and linking cycles in surgery theory, 1974
2. ^ John W. Morgan, A product formula for surgery obstructions, 1978
3. ^ (Lusztig, Milnor & Peterson 1969)

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